Exercise 1.1 - Class Notes

1. Determine whether each of the following relations are reflexive, symmetric and transitive.

(i) Relation $\text{R}$ in the set $\text{A = }\left\{ \text{1, 2, 3}...\text{13, 14} \right\}$ defined as $\text{R = }\left\{ \left( \text{x, y} \right)\text{: 3x - y = 0} \right\}$

Ans: The given relation is: $\text{R = }\left\{ \left( \text{1, 3} \right)\text{, }\left( \text{2, 6} \right)\text{, }\left( \text{3, 9} \right)\text{, }\left( \text{4, 12} \right) \right\}$ Since $\left( \text{1, 1} \right)\text{, }\left( \text{2, 2} \right)\text{ }...$ and $\left( \text{14, 14} \right)\notin R$. We conclude that $\text{R}$ is not reflexive. Since $\left( \text{1, 3} \right)\in \text{R}$, but $\left( \text{3, 1} \right)\notin \text{R}$. since $\text{3}\left( \text{3} \right)\text{-1}\ne \text{0}$ We conclude that $\text{R}$ is not symmetric. Since $\left( \text{1, 3} \right)$ and $\left( \text{3, 9} \right)\in \text{R}$, but$\left( \text{1, 9} \right)\notin \text{R}\text{. }\left[ \text{3}\left( \text{1} \right)\text{-9}\ne \text{0} \right]$. We conclude that $\text{R}$ is not transitive. Therefore, the relation $\text{R}$ is not reflexive, symmetric or transitive.

Relation $\text{R}$ in the set $\text{N}$ of natural numbers defined as $\text{R = }\!\!\{\!\!\text{ }\left( \text{x, y} \right)\text{: y = x + 5}$ and $\text{x<4 }\!\!\}\!\!\text{ }$
Ans: The given relation is: $\text{R = }\left\{ \left( \text{1, 6} \right)\text{, }\left( \text{2, 7} \right)\text{, }\left( \text{3, 8} \right) \right\}$. Since $\left( \text{1, 1} \right)\notin \text{R}$. We conclude that $\text{R}$ is not reflexive. Since $\left( \text{1, 6} \right)\in \text{R}$ but $\left( \text{6, 1} \right)\notin \text{R}$. We conclude that $\text{R}$ is not symmetric. In the given relation $\text{R}$ there is not any ordered pair such that $\left( \text{x, y} \right)$ and $\left( \text{y, z} \right)$ both $\in \text{R}$, therefore we can say that $\left( \text{x, z} \right)$ cannot belong to $\text{R}$. Therefore $\text{R}$ is not tr
Ansitive. Hence, the given relation $\text{R}$ is not reflexive, symmetric or tr
Ansitive.

Relation $\text{R}$in the set $\text{A = }\!\!\{\!\!\text{ 1, 2, 3, 4, 5, 6 }\!\!\}\!\!\text{ }$ as $\text{R= }\!\!\{\!\!\text{ }\left( \text{x, y} \right)\text{: y}$is divisible by $\text{x }\!\!\}\!\!\text{ }$
Ans: The given relation is $\text{R = }\!\!\{\!\!\text{ }\left( \text{x, y} \right)\text{ : y}$ is divisible by $\text{x }\!\!\}\!\!\text{ }$ As we know that any number except $\text{0}$ is divisible by itself, therefore $\left( \text{x, x} \right)\in \text{R}$. We conclude that $\text{R}$ is reflexive. Since $\left( \text{2, 4} \right)\in \text{R}$ (because 4 is divisible by 2, but $\left( \text{4, 2} \right)\notin \text{R}$ (since 2 is not divisible by 4. We conclude that $\text{R}$ is not symmetric. Assuming that $\left( \text{x, y} \right)$ and $\left( \text{y, z} \right)\in \text{R}$, $\text{y}$ is divisible by $\text{x}$ and $\text{z}$ is divisible by $\text{y}$. Hence $\text{z}$ is divisible by $\text{x}\Rightarrow \left( \text{x, z} \right)\in \text{R}$. We conclude that $\text{R}$ is tr
Ansitive. Hence, the given relation $\text{R}$ is reflexive and tr
Ansitive but it is not symmetric.

Relation $\text{R}$ in the set $\text{Z}$ of all integers defined as $\text{R = }\!\!\{\!\!\text{ }\left( \text{x, y} \right)\text{: x - y }\!\!\}\!\!\text{ }$ is as integer
Ans: The given relation is $\text{R = }\!\!\{\!\!\text{ }\left( \text{x, y} \right)\text{: x - y}$ is an integer$\text{ }\!\!\}\!\!\text{ }$ If $\text{x}\in \text{Z, }\left( \text{x, x} \right)\in \text{R}$ because $\text{x-x = 0}$ is an integer. Hence, we conclude that $\text{R}$ is reflexive. For $\text{x, y}\in \text{Z}$, if $\left( \text{x, y} \right)\in \text{R}$, then $\text{x - y}$ is an integer and therefore $\left( \text{y-x} \right)$ is also an integer. Therefore, we conclude that $\left( \text{y, x} \right)\in \text{R}$and hence $\text{R}$ is symmetric. Assuming that $\left( \text{x, y} \right)$ and $\left( \text{y, z} \right)\in \text{R}$, where $\text{x, y, z}\in \text{Z}$. We can say that $\left( \text{x-y} \right)$ and $\left( \text{y-z} \right)$ are integers. so, $\left( \text{x, z} \right)\in \text{R}$ Hence, we conclude that $\text{R}$ is tr
Ansitive. Therefore the given relation $\text{R}$ is reflexive, symmetric, and tr
Ansitive.

Relation $\text{R}$ in the set $\text{A}$ of human beings in a town at a particular time given by The relation is: $\text{R = }\!\!\{\!\!\text{ }\left( \text{x, y} \right)\text{: x and y}$ work at the same place$\}$
Ans: The given relation is: $\text{R = }\!\!\{\!\!\text{ }\left( \text{x, y} \right)\text{: x}$ and $\text{y}$ work at the same place$\text{ }\!\!\}\!\!\text{ }$ This implies that $\left( \text{x, x} \right)\in \text{R}$. Hence, we conclude that $\text{R}$ is reflexive. Now, $\left( \text{x, y} \right)\in \text{R}$, then $\text{x}$ and $\text{y}$ work at the same place, which me
Ans $\text{y}$ and $\text{x}$ also work at the same place. Therefore, $\left( \text{y, x} \right)\in \text{R}$. Hence, we conclude that $\text{R}$ is symmetric. Let us assume that $\left( \text{x, y} \right)\text{, }\left( \text{y, z} \right)\in \text{R}$. Then, we can say that $\text{x}$ and $\text{y}$ work at the same place and $\text{y}$ and $\text{z}$ work at the same place. Which me
Ans that $\text{x}$ and $\text{z}$ also work at the same place. Therefore, $\left( \text{x, z} \right)\in \text{R}$. Hence, we conclude that $\text{R}$ is tr
Ansitive. Therefore, the given relation $\text{R}$ is reflexive, symmetric and tr
Ansitive.

The relation is: $\text{R = }\!\!\{\!\!\text{ }\left( \text{x, y} \right)\text{: x and y}$ live in the same locality$\text{ }\!\!\}\!\!\text{ }$
Ans: The given relation is $\text{R = }\!\!\{\!\!\text{ }\left( \text{x, y} \right)\text{: x and y}$ live in the same locality$\text{ }\!\!\}\!\!\text{ }$ Since, $\left( \text{x, x} \right)\in \text{R}$. Therefore, we conclude that $\text{R}$ is reflexive. Since $\left( \text{x, y} \right)\in \text{R}$, $\text{x}$ and $\text{y}$ live in the same locality. Therefore, $\text{y}$ and $\text{x}$ also live in the same locality, so, $\left( \text{y, x} \right)\in \text{R}$. Hence, $\text{R}$ is symmetric. Let $\left( \text{x, y} \right)\in \text{R}$ and $\left( \text{y, z} \right)\in \text{R}$. Hence $\text{x}$ and $\text{y}$ live in the same locality and $\text{y}$ and $\text{z}$ also live in the same locality. Which me
Ans that $\text{x}$ and $\text{z}$ also live in the same locality. Therefore, $\left( \text{x, z} \right)\in \text{R}$. Hence, we conclude that $\text{R}$ is tr
Ansitive. Therefore, the given relation $\text{R}$ is reflexive, symmetric and tr
Ansitive.

$\text{R = }\!\!\{\!\!\text{ }\left( \text{x, y} \right)\text{: x}$ is exactly $\text{7}$cm taller than $\text{y }\!\!\}\!\!\text{ }$
Ans: The given relation is: $\text{R = }\!\!\{\!\!\text{ }\left( \text{x, y} \right)\text{: x}$ is exactly $\text{7}$ cm taller than $\text{y }\!\!\}\!\!\text{ }$ Since, $\left( \text{x, x} \right)\notin \text{R}$. Therefore, we conclude that $\text{R}$ is not reflexive. Let $\left( \text{x, y} \right)\in \text{R}$ , Since $\text{x}$ is exactly $\text{7}$ cm taller than $\text{y}$, therefore $\text{y}$ is obviously not taller than $\text{x}$, so, $\left( \text{y, x} \right)\notin \text{R}$. Hence, $\text{R}$ is not symmetric. Assuming that $\left( \text{x, y} \right)\text{, }\left( \text{y, z} \right)\in \text{R}$, we can say that $\text{x}$ is exactly $\text{7}$ cm taller than $\text{y}$and $\text{y}$ is exactly $\text{7}$ cm taller than $\text{z}$. Which me
Ans that $\text{x}$ is exactly $\text{14}$ cm taller than $\text{z}$. So, $\left( \text{x, z} \right)\notin \text{R}$. Hence, $\text{R}$ is not tr
Ansitive. Therefore, the given relation $\text{R}$ is not reflexive, symmetric or tr
Ansitive.

$\text{R = }\!\!\{\!\!\text{ }\left( \text{x, y} \right)\text{: x}$ is wife of $\text{y }\!\!\}\!\!\text{ }$
Ans: The given relation is: $\text{R = }\!\!\{\!\!\text{ }\left( \text{x, y} \right)\text{: x}$ is the wife of $\text{y }\!\!\}\!\!\text{ }$. Since, $\left( \text{x, x} \right)\notin \text{R}$ Therefore, we conclude that $\text{R}$ is not reflexive. Let $\left( \text{x, y} \right)\in \text{R}$ , Since $\text{x}$ is the wife of $\text{y}$, therefore $\text{y}$ is obviously not the wife of $\text{x}$, so, $\left( \text{y, x} \right)\notin \text{R}$. Hence, $\text{R}$ is not symmetric. Assuming that $\left( \text{x, y} \right)\text{, }\left( \text{y, z} \right)\in \text{R}$, we can say that $\text{x}$ is the wife of $\text{y}$and $\text{y}$ is the wife of $\text{z}$, which is not possible. So, $\left( \text{x, z} \right)\notin \text{R}$. Hence, $\text{R}$ is not tr
Ansitive. Therefore the given relation $\text{R}$is not reflexive, symmetric or tr
Ansitive.

$\text{R = }\!\!\{\!\!\text{ }\left( \text{x, y} \right)\text{: x}$ is father of $\text{y }\!\!\}\!\!\text{ }$
Ans: The given relation is: $\text{R = }\!\!\{\!\!\text{ }\left( \text{x, y} \right)\text{: x}$ is the father of $\text{y }\!\!\}\!\!\text{ }$ Since, $\left( \text{x, x} \right)\notin \text{R}$ Therefore, we conclude that $\text{R}$ is not reflexive. Let $\left( \text{x, y} \right)\in \text{R}$ , Since $\text{x}$ is the father of $\text{y}$, therefore $\text{y}$ is obviously not the father of $\text{x}$, so, $\left( \text{y, x} \right)\notin \text{R}$. Hence, $\text{R}$ is not symmetric. Assuming that $\left( \text{x, y} \right)\text{, }\left( \text{y, z} \right)\in \text{R}$, we can say that $\text{x}$ is the father of $\text{y}$and $\text{y}$ is the father of $\text{z}$, then $\text{x}$ is not the father of $\text{z}$. So, $\left( \text{x, z} \right)\notin \text{R}$. Hence, $\text{R}$ is not tr
Ansitive. Therefore the given relation $\text{R}$is not reflexive, symmetric or tr
Ansitive.

2. Show that the relation $\text{R}$ in the set $\text{R}$ of real numbers, defined $\text{R = }\left\{ \left( \text{a, b} \right)\text{: a}\le {{\text{b}}^{\text{2}}} \right\}$ is neither reflexive nor symmetric nor tr
Ansitive.
Ans: The given relation is: $\text{R = }\left\{ \left( \text{a, b} \right)\text{: a}\le {{\text{b}}^{\text{2}}} \right\}$ Since $\left( \dfrac{\text{1}}{\text{2}}\text{, }\dfrac{\text{1}}{\text{2}} \right)\notin \text{R}$. (Since $\frac{1}{2}$ is not less than $\frac{1}{4}$) Therefore, $\text{R}$ is not reflexive. Since $\left( \text{1, 4} \right)\in \text{R}$ as $\text{1}{{\text{4}}^{\text{2}}}$, but $\left( \text{4, 1} \right)\notin \text{R}$ as ${{\text{4}}^{2}}$ is not less than ${{\text{1}}^{2}}$. Therefore $\text{R}$ is not symmetric. Assuming that $\left( \text{3, 2} \right)\text{, }\left( \text{2, 1}\text{.5} \right)\in \text{R}$, so, $\text{3}{{\text{2}}^{\text{2}}}\text{=4}$ and $\text{2}{{\left( \text{1}\text{.5} \right)}^{\text{2}}}\text{=2}\text{.25}$ but $\text{3}{{\left( \text{1}\text{.5} \right)}^{\text{2}}}\text{=2}\text{.25}$. Hence, $\text{R}$ is not tr
Ansitive. Therefore, the given relation $\text{R}$ is neither reflexive, nor symmetric, nor tr
Ansitive.

3. Check whether the relation $\text{R}$ defined in the set $\left\{ \text{1, 2, 3, 4, 5, 6} \right\}$ as $\text{R = }\left\{ \left( \text{a, b} \right)\text{: b=a+1} \right\}$ is reflexive, symmetric or tr
Ansitive.
Ans: The given relation is $\text{R = }\left\{ \left( \text{a, b} \right)\text{: b=a+1} \right\}$ defined in the set $\text{A=}\left\{ \text{1, 2, 3, 4, 5, 6} \right\}$. So, $\text{R=}\left\{ \left( \text{1, 2} \right)\text{, }\left( \text{2, 3} \right)\text{, }\left( \text{3, 4} \right)\text{, }\left( \text{4, 5} \right)\text{, }\left( \text{5, 6} \right) \right\}$ Since, $\left( \text{a, a} \right)\notin \text{R,a}\in \text{A}$. $\left( \text{1, 1} \right)\text{, }\left( \text{2, 2} \right)\text{, }\left( \text{3, 3} \right)\text{, }\left( \text{4, 4} \right)\text{, }\left( \text{5, 5} \right)\text{, }\left( \text{6, 6} \right)\notin \text{R}$ Therefore, $\text{R}$ is not reflexive Since, $\left( \text{1, 2} \right)\in \text{R}$, but $\left( \text{2, 1} \right)\notin \text{R}$. Therefore $\text{R}$ is not symmetric. Since $\left( \text{1, 2} \right)\text{, }\left( \text{2, 3} \right)\in \text{R}$, but $\left( \text{1, 3} \right)\notin \text{R}$. Hence, $\text{R}$ is not tr
Ansitive. Therefore, the given relation $\text{R}$ is neither reflexive, nor symmetric, nor tr
Ansitive

4. Show that the relation $\text{R}$ in $\text{R}$ defined as $\text{R = }\left\{ \left( \text{a, b} \right)\text{: a}\le \text{b} \right\}$ is reflexive and tr
Ansitive but not symmetric.
Ans: The given relation is $\text{R = }\left\{ \left( \text{a, b} \right)\text{: a}\le \text{b} \right\}$. Since, $\left( \text{a, a} \right)\in R$. Therefore, $\text{R}$ is reflexive. Since, $\left( \text{2, 4} \right)\in R$ (as $\text{2,4}$), but $\left( \text{4, 2} \right)\notin R$(as $\text{4,2}$). Therefore $\text{R}$ is not symmetric. Assuming that $\left( \text{a, b} \right)\text{, }\left( \text{b, c} \right)\in R$, $\text{a}\le \text{b}$ and $b\le c$, therefore, $a\le c$. Hence, $\text{R}$ is tr
Ansitive. Therefore, the given relation $\text{R}$ is reflexive and tr
Ansitive but not symmetric.

5. Check whether the relation $\text{R}$ in $\text{R}$ defined as $\text{R = }\left\{ \left( \text{a, b} \right)\text{: a}\le {{\text{b}}^{\text{3}}} \right\}$ is reflexive, symmetric or tr
Ansitive.
Ans: The given relation is: $\text{R = }\left\{ \left( \text{a, b} \right)\text{: a}\le {{\text{b}}^{\text{3}}} \right\}$ Since $\left( \dfrac{\text{1}}{\text{2}}\text{, }\dfrac{\text{1}}{8} \right)\notin \text{R}$. (Since $\frac{1}{2}$ is not less than $\frac{1}{8}$) Therefore, $\text{R}$ is not reflexive. Since $\left( \text{1, 4} \right)\in \text{R}$ as $\text{1}{{\text{4}}^{\text{3}}}$, but $\left( \text{4, 1} \right)\notin \text{R}$ as $\text{4}$ is not less than ${{\text{1}}^{\text{3}}}$. Therefore $\text{R}$ is not symmetric. Assuming that $\left( \text{3, }\dfrac{\text{3}}{\text{2}} \right)\text{, }\left( \dfrac{\text{3}}{\text{2}}\text{, }\dfrac{\text{6}}{\text{5}} \right)\in \text{R}$, so, $\text{3}{{\left( \dfrac{\text{3}}{\text{2}} \right)}^{\text{3}}}$ and $\dfrac{\text{3}}{\text{2}}\text{}{{\left( \dfrac{\text{6}}{\text{5}} \right)}^{\text{3}}}$ but $\text{3}{{\left( \dfrac{\text{6}}{\text{5}} \right)}^{\text{3}}}\notin R$. Hence, $\text{R}$ is not tr
Ansitive. Therefore, the given relation $\text{R}$ is neither reflexive, nor symmetric, nor tr
Ansitive.

6. Show that the relation $\text{R}$ in the set $\left\{ \text{1, 2, 3} \right\}$ given by $\text{R = }\left\{ \left( \text{1, 2} \right)\text{, }\left( \text{2, 1} \right) \right\}$ is symmetric but neither reflexive nor tr
Ansitive.
Ans: The given relation is $\text{R = }\left\{ \left( \text{1, 2} \right)\text{, }\left( \text{2, 1} \right) \right\}$ on the set $\text{A=}\left\{ \text{1, 2, 3} \right\}$. Since $\left( \text{1, 1} \right)\text{, }\left( \text{2, 2} \right)\text{, }\left( \text{3, 3} \right)\notin \text{R}$ Therefore, $\text{R}$ is not reflexive. Since, $\left( \text{1, 2} \right)\in R$ and $\left( \text{2, 1} \right)\in R$. Therefore $\text{R}$ is symmetric. Since, $\left( \text{1, 2} \right)\in R$ and $\left( \text{2, 1} \right)\in R$, but $\left( \text{1, 1} \right)\notin \text{R}$. Hence, $\text{R}$ is not tr
Ansitive. Therefore, the given relation $\text{R}$ is symmetric but neither reflexive nor tr
Ansitive.

7. Show that the relation $\text{R}$ in the set $\text{A}$ of all the books in a library of a college, given by $\text{R = }\!\!\{\!\!\text{ }\left( \text{x,}\,\,\text{y} \right)\text{: x}$ and $\text{y}$ have same number of pages$\text{ }\!\!\}\!\!\text{ }$ is an equivalence relation.
Ans: The given relation is: $\text{R = }\!\!\{\!\!\text{ }\left( \text{x,}\,\,\text{y} \right)\text{: x}$ and $\text{y}$ have the same number of pages$\text{ }\!\!\}\!\!\text{ }$ Since $\left( \text{x, x} \right)\in \text{R}$ as $\text{x}$ and $\text{x}$ have the same number of pages. Therefore, $\text{R}$ is reflexive. Let $\left( \text{x, y} \right)\in R$, so $\text{x}$ and $\text{y}$ have the same number of pages, $\left( \text{y, x} \right)\in \text{R}$ therefore $\text{y}$ and $\text{x}$ will also have the same number of pages. Therefore $\text{R}$ is symmetric. Assuming $\left( \text{x, y} \right)\in R$ and $\left( \text{y, z} \right)\in R$. $\text{x}$ and $\text{y}$ have the same number of pages and $\text{y}$ and $\text{z}$ also have the same number of pages. Therefore, $\text{x}$ and $\text{z}$ will also have the same number of pages. So, $\left( \text{x, z} \right)\in R$. Hence, $\text{R}$ is tr
Ansitive. Therefore, the given relation $\text{R}$ is an equivalence relation.

8. Show that the relation $\text{R}$ in the set $\text{A=}\left\{ \text{1, 2, }\!\!~\!\!\text{ 3, 4, 5} \right\}$ given by $\text{R = }\!\!\{\!\!\text{ }\left( \text{a, b} \right)\text{: }\left| \text{a-b} \right|$ is even$\text{ }\!\!\}\!\!\text{ }$, is an equivalence relation. Show that all the elements of $\left\{ \text{1, }\!\!~\!\!\text{ 3, 5} \right\}$ are related to each other and all the elements of $\left\{ \text{2, 4} \right\}$ are related to each other. But no element of $\left\{ \text{1, }\!\!~\!\!\text{ 3, 5} \right\}$ is related to any element of $\left\{ \text{2, 4} \right\}$.
Ans: Let $\text{a}\in \text{A}$, So, $\left| \text{a-a} \right|\text{ = 0}$ (which is an even number). Therefore, $\text{R}$ is reflexive. Let $\left( \text{a, b} \right)\in \text{R}$, Now, $\left| \text{a-b} \right|$ is even, Hence $\left| \text{a-b} \right|$ and $\left| \text{b-a} \right|$ are both even Therefore, $\left( \text{b, a} \right)\in \text{R}$ Therefore $\text{R}$ is symmetric Let $\left( \text{a, b} \right)\in \text{R}$and $\left( \text{b, c} \right)\in \text{R}$, $\Rightarrow \left| \text{a-b} \right|$ is even and $\left| \text{b-c} \right|$ is even $\Rightarrow \left| \text{a-c} \right|$ is even. $\Rightarrow \left( \text{a, c} \right)\in \text{R}$ Therefore, $\text{R}$ is tr
Ansitive. Therefore, the given relation $\text{R}$ is an equivalence relation. All the elements of the set $\left\{ \text{1, }\!\!~\!\!\text{ 3, 5} \right\}$ are all odd. Hence, the modulus of the difference of any two elements will be an even number. So, all the elements of this set are related to each other. All elements of $\left\{ \text{2, 4} \right\}$ are even while all the elements of $\left\{ \text{1, }\!\!~\!\!\text{ 3, 5} \right\}$ are odd so no element of $\left\{ \text{1, }\!\!~\!\!\text{ 3, 5} \right\}$ can be related to any element of$\left\{ 2,\text{ }4 \right\}$. Therefore, the absolute value of the difference between the two elements (from each of these two subsets) will not be an even value.

9. Show that each of the relation $\text{R}$ in the set $\text{A = }\!\!\{\!\!\text{ x}\in \text{Z: 0}\le \text{x}\le \text{12 }\!\!\}\!\!\text{ }$ , is an equivalence relation. Find the set of all elements related to 1 in each case. $\text{R = }\!\!\{\!\!\text{ }\left( \text{a, b} \right)\text{ : }\left| \text{a -- b} \right|$ is a multiple of $\text{4 }\!\!\}\!\!\text{ }$
Ans: The given set $\text{A = }\!\!\{\!\!\text{ x}\in \text{Z: 0}\le \text{x}\le \text{12 }\!\!\}\!\!\text{ = }\left\{ \text{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} \right\}$ The given relation is: $\text{R = }\!\!\{\!\!\text{ }\left( \text{a, b} \right)\text{ : }\left| \text{a -- b} \right|$ is a multiple of $\text{4 }\!\!\}\!\!\text{ }$. Let $a\in A$, $\left( \text{a, a} \right)\in R$ as $\left| \text{a-a} \right|\text{=0}$ is a multiple of $\text{4}$. Therefore, $\text{R}$ is reflexive. Let, $\left( \text{a, b} \right)\in \text{R}\Rightarrow \left| \text{a-b} \right|$ is a multiple of $\text{4}$. $\Rightarrow \left| \text{-}\left( \text{a-b} \right) \right|\text{=}\left| \text{b-a} \right|$ is a multiple of $\text{4}$. $\Rightarrow \left( \text{b, a} \right)\in \text{R}$ Therefore $\text{R}$ is symmetric. $\left( \text{a, b} \right)\text{, }\left( \text{b, c} \right)\in R$. $\Rightarrow \left| \text{a-b} \right|$ is a multiple of $\text{4}$ and $\left| \text{b-c} \right|$ is a multiple of $\text{4}$. $\Rightarrow \left( \text{a-b} \right)$ is a multiple of $\text{4}$ and $\left( \text{b-c} \right)$ is a multiple of $\text{4}$. $\Rightarrow \left( \text{a-c} \right)\text{=}\left( \text{a-b} \right)\text{+}\left( \text{b-c} \right)$ is a multiple of $\text{4}$. $\Rightarrow \left| \text{a-c} \right|$ is a multiple of $\text{4}$. $\Rightarrow \left( \text{a, c} \right)\in \text{R}$ Therefore, $\text{R}$ is tr
Ansitive. Therefore, the given relation $\text{R}$ is an equivalence relation. The set of elements related to 1 is $\left\{ \text{1, }\!\!~\!\!\text{ 5, 9} \right\}$

$\text{ }\!\!~\!\!\text{ R = }\left\{ \left( \text{a, b} \right)\text{ : a = b} \right\}$
Ans: The given relation is: $\text{R = }\left\{ \left( \text{a, b} \right)\text{ : a = b} \right\}$. $a\in A,\left( \text{a, a} \right)\in R$, since $\text{a = a}$. Therefore, $\text{R}$ is reflexive. Let $\left( \text{a, b} \right)\in \text{R}\Rightarrow \text{a=b}$. $\Rightarrow b\text{=a}\Rightarrow \left( \text{b, a} \right)\in R$ Therefore $\text{R}$ is symmetric. $\left( \text{a, b} \right)\text{, }\left( \text{b, c} \right)\in R$ $\Rightarrow a\text{=b}$ and $b\text{=c}$ $\Rightarrow a\text{=c}$ $\Rightarrow \left( \text{a, c} \right)\in R$ Therefore, $\text{R}$ is tr
Ansitive. Therefore, the given relation $\text{R}$ is an equivalence relation. The set of elements related to $\text{1}$ is $\left\{ \text{1} \right\}$.

10. Give an example of a relation. Which is Symmetric but neither reflexive nor tr
Ansitive.
Ans: Let us assume the relation $\text{R= }\left\{ \left( \text{5, 6} \right)\text{, }\left( \text{6, 5} \right) \right\}$ in set $\text{A= }\left\{ \text{5, 6, 7} \right\}$. So, the relation $\text{R}$ is not reflexive as $\left( \text{5, 5} \right)\text{, }\left( \text{6, 6} \right)\text{, }\left( \text{7, 7} \right)\notin \text{R}$. The relation $\text{R}$ is symmetric as $\left( \text{5, 6} \right)\in R$ and $\left( \text{6, 5} \right)\in R$. The relation $\text{R}$ is not tr
Ansitive as $\left( \text{5, 6} \right)\text{, }\left( \text{6, 5} \right)\in R$ , but $\left( \text{5, 5} \right)\notin \text{R}$. Therefore, the given relation $\text{R}$ is symmetric but not reflexive or tr
Ansitive.

Tr
Ansitive but neither reflexive nor symmetric.
Ans: Let us assume the relation $\text{R = }\left\{ \left( \text{a, b} \right)\text{ : a b} \right\}$ So, the relation $\text{R}$ is not reflexive because for $a\in R$, $\left( \text{a, a} \right)\notin \text{R}$ since a cannot be strictly less than itself. Let $\left( {1,2} \right) \in R\left( {as1 < 2} \right)$ Since $\text{2}$ is not less than $\text{1}$, $\left( \text{2, 1} \right)\notin \text{R}$. Therefore $\text{R}$ is not symmetric. Let $\left( \text{a, b} \right)\text{, }\left( \text{b, c} \right)\in R$. $\Rightarrow \left( \text{a, c} \right)\in R$ Therefore, $\text{R}$ is tr
Ansitive. So, the relation $\text{R}$ is tr
Ansitive but not reflexive and symmetric.

Reflexive and symmetric but not tr
Ansitive.
Ans: Let us assume the relation $R=\left\{ \left( \text{4, 4} \right)\text{, }\left( \text{6, 6} \right)\text{, }\left( \text{8, 8} \right),\text{ }\left( \text{4, 6} \right),\text{ }\left( \text{6, 4} \right),\text{ }\left( \text{6, 8} \right),\text{ }\left( \text{8, 6} \right) \right\}$ in set $\text{A= }\left\{ \text{4, 6, 8} \right\}$. The relation $\text{R}$ is reflexive since for $a\in R$, $\left( \text{a, a} \right)\in R$. The relation $\text{R}$ is symmetric since $\left( \text{a, b} \right)\in R\Rightarrow \left( \text{b, a} \right)\in R$ for $a,b\in R$. The relation $\text{R}$ is not tr
Ansitive since $\left( \text{4, 6} \right)\text{, }\left( \text{6, 8} \right)\in R$, but $\left( \text{4, 8} \right)\notin \text{R}$. Therefore the relation $\text{R}$ is reflexive and symmetric but not tr
Ansitive.

Reflexive and tr
Ansitive but not symmetric.
Ans: Let us take the relation $\text{ }\!\!~\!\!\text{ R = }\left\{ \left( \text{a, b} \right)\text{ : }{{\text{a}}^{\text{3}}}\ge {{\text{b}}^{\text{3}}} \right\}$. Since $\left( \text{a, b} \right)\in \text{R}$. Therefore $\text{R}$ is reflexive. Since $\left( \text{2, 1} \right)\in \text{R}$, but $\left( \text{1, 2} \right)\notin \text{R}$, Therefore $\text{R}$ is not symmetric. Let $\left( {a,b} \right),\left( {b,c} \right) \in R$ $\Rightarrow {{\text{a}}^{\text{3}}}\ge {{\text{b}}^{\text{3}}}$ and ${{\text{b}}^{\text{3}}}\ge {{\text{c}}^{\text{3}}}$ $\Rightarrow {{\text{a}}^{\text{3}}}\ge {{\text{c}}^{\text{3}}}$ $\Rightarrow \left( \text{a, c} \right)\in \text{R}$ Therefore $\text{R}$ is tr
Ansitive. Therefore the relation $\text{R}$ is reflexive and tr
Ansitive but not symmetric.

Symmetric and tr
Ansitive but not reflexive.
Ans: Let us take a relation $\text{R=}\left\{ \left( \text{-5, -6} \right)\text{, }\left( \text{-6, -5} \right)\text{, }\left( \text{-5, -5} \right) \right\}$ in set $\text{A=}\left\{ \text{-5, -6} \right\}$. The relation $\text{R}$ is not reflexive as $\left( \text{-6, -6} \right)\notin \text{R}$. Since $\left( \text{-5, -6} \right)\in \text{R}$ and $\left( \text{-6, -5} \right)\in \text{R}$. Therefore $\text{R}$ is symmetric. Since $\left( \text{-5, -6} \right)\text{, }\left( \text{-6, -5} \right)\in \text{R}$ and $\left( \text{-5, -5} \right)\in \text{R}$. Therefore $\text{R}$ is tr
Ansitive. Therefore the relation $\text{R}$ is symmetric and tr
Ansitive but not reflexive.

11. Show that the relation $\text{R}$ in the set $\text{A}$ of points in a plane given by $\text{R = }\!\!\{\!\!\text{ }\left( \text{P, Q} \right)\text{ :}$ Distance of the point $\text{P}$ from the origin is same as the distance of the point $\text{Q}$ from the origin}, is an equivalence relation. Further, show that the set of all points related to a point $\text{P}\ne \left( \text{0, 0} \right)$ is the circle passing through $\text{P}$ with origin as centre.
Ans: The given relation is $\text{ }\!\!~\!\!\text{ R = }\!\!\{\!\!\text{ }\left( \text{P, Q} \right)\text{ :}$ Distance of $\text{P}$ from the origin is the same as the distance of $\text{Q}$ from the origin} Since, $\left( \text{P, P} \right)\in \text{R}$. The relation $\text{R}$ is reflexive. Let $\left( \text{P, Q} \right)\in \text{R}$, distance of $\text{P}$ from the origin is the same as the distance of $\text{Q}$ from the origin similarly distance of $\text{Q}$ from the origin will be the same as the distance of $\text{P}$ from the origin. So, $\left( \text{Q, P} \right)\in \text{R}$. Therefore $\text{R}$ is symmetric. Let $\left( \text{P, Q} \right),\left( \text{Q, S} \right)\in \text{R}$. Distance of $\text{P}$ from the origin is the same as the distance of $\text{Q}$ from the origin and distance of $\text{Q}$ from the origin is the same as the distance of $\text{S}$ from the origin. So, the distance of $\text{S}$ from the origin will be the same as the distance of $\text{P}$ from the origin. So, $\left( \text{P, S} \right)\in \text{R}$. Therefore $\text{R}$ is tr
Ansitive. Therefore the relation $\text{R}$ is an equivalence relation. The set of points related to $\text{P}\ne \left( \text{0, 0} \right)$ will be those points whose distance from origin is same as distance of $\text{P}$ from the origin and will form a circle with the centre as origin and this circle passes through $\text{P}$.

12. Show that the relation $\text{R}$ is defined in the set $\text{A}$ of all triangles as $\text{R = }\!\!\{\!\!\text{ }\left( {{\text{T}}_{\text{1}}}\text{, }{{\text{T}}_{\text{2}}} \right)\text{ : }{{\text{T}}_{\text{1}}}$ is similar to ${{\text{T}}_{\text{2}}}\text{ }\!\!\}\!\!\text{ }$, is equivalence relation. Consider three right angle triangles ${{\text{T}}_{\text{1}}}$ with sides $\text{3, 4, 5}$ and ${{\text{T}}_{\text{2}}}$ with sides $\text{5, 12, 13}$ and ${{\text{T}}_{\text{3}}}$ with sides $\text{6, 8, 10}$. Which triangles among ${{\text{T}}_{\text{1}}}\text{, }{{\text{T}}_{\text{2}}}$ and ${{\text{T}}_{\text{3}}}$ are related?
Ans: The given relation is $\text{R = }\!\!\{\!\!\text{ }\left( {{\text{T}}_{\text{1}}}\text{, }{{\text{T}}_{\text{2}}} \right)\text{ : }{{\text{T}}_{\text{1}}}$ is similar to ${{\text{T}}_{\text{2}}}\text{ }\!\!\}\!\!\text{ }$ since, every triangle is similar to itself. The relation $\text{R}$ is reflexive. If $\left( {{\text{T}}_{\text{1}}}\text{, }{{\text{T}}_{\text{2}}} \right)\in \text{R}$, then ${{\text{T}}_{\text{1}}}$ is similar to ${{\text{T}}_{\text{2}}}$. $\Rightarrow {{\text{T}}_{\text{2}}}$ is similar to ${{\text{T}}_{\text{1}}}$. $\Rightarrow \left( {{\text{T}}_{\text{2}}}\text{, }{{\text{T}}_{\text{1}}} \right)\in \text{R}$ Therefore $\text{R}$ is symmetric. Let $\left( {{\text{T}}_{\text{1}}}\text{, }{{\text{T}}_{\text{2}}} \right)\text{,}\left( {{\text{T}}_{\text{2}}}\text{, }{{\text{T}}_{\text{3}}} \right)\in \text{R}$. $\Rightarrow {{\text{T}}_{\text{1}}}$ is similar to ${{\text{T}}_{\text{2}}}$ and ${{\text{T}}_{\text{2}}}$ is similar to ${{\text{T}}_{\text{3}}}$. $\Rightarrow {{\text{T}}_{\text{1}}}$ is similar to ${{\text{T}}_{\text{3}}}$. $\Rightarrow \left( {{\text{T}}_{\text{1}}}\text{, }{{\text{T}}_{\text{3}}} \right)\in \text{R}$ Therefore $\text{R}$ is tr
Ansitive. Therefore the relation $\text{R}$ is an equivalence relation. $\dfrac{\text{3}}{\text{6}}\text{=}\dfrac{\text{4}}{\text{8}}\text{=}\dfrac{\text{5}}{\text{10}}\left( \text{=}\dfrac{\text{1}}{\text{2}} \right)$ Since, the corresponding sides of triangles ${{\text{T}}_{\text{1}}}$ and ${{\text{T}}_{\text{3}}}$ are in the same ratio, therefore triangle ${{\text{T}}_{\text{1}}}$ is similar to triangle ${{\text{T}}_{\text{3}}}$. Hence, ${{\text{T}}_{\text{1}}}$ is related to ${{\text{T}}_{\text{3}}}$.

13. Show that the relation $\text{R}$ defined in the set $\text{A}$ of all polygons as $\text{R = }\!\!\{\!\!\text{ }\left( {{\text{P}}_{\text{1}}}\text{, }{{\text{P}}_{\text{2}}} \right)\text{: }{{\text{P}}_{\text{1}}}$ and ${{\text{P}}_{\text{2}}}$ have same number of sides$\text{ }\!\!\}\!\!\text{ }$, is an equivalence relation. What is the set of all elements in $\text{A}$ related to the right angle triangle $\text{T}$ with sides $\text{3, 4}$ and $\text{5}$?
Ans: $\text{R = }\!\!\{\!\!\text{ }\left( {{\text{P}}_{\text{1}}}\text{, }{{\text{P}}_{\text{2}}} \right)\text{: }{{\text{P}}_{\text{1}}}$ and ${{\text{P}}_{\text{2}}}$ have same number of sides$\text{ }\!\!\}\!\!\text{ }$. Since $\left( {{\text{P}}_{\text{1}}}\text{, }{{\text{P}}_{\text{1}}} \right)\in \text{R}$ , as same polygon has same number of sides. The relation $\text{R}$ is reflexive. Let $\left( {{\text{P}}_{\text{1}}}\text{, }{{\text{P}}_{\text{2}}} \right)\in \text{R}$ . $\Rightarrow {{\text{P}}_{\text{1}}}$ and ${{\text{P}}_{\text{2}}}$ have same number of sides. $\Rightarrow {{\text{P}}_{\text{2}}}$ and ${{\text{P}}_{\text{1}}}$ have same number of sides. $\Rightarrow \left( {{\text{P}}_{\text{2}}}\text{, }{{\text{P}}_{\text{1}}} \right)\in \text{R}$ Therefore $\text{R}$ is symmetric. Let $\left( {{\text{P}}_{\text{1}}}\text{, }{{\text{P}}_{\text{2}}} \right)\text{,}\left( {{\text{P}}_{\text{2}}}\text{, }{{\text{P}}_{\text{3}}} \right)\in \text{R}$. $\Rightarrow {{\text{P}}_{\text{1}}}$ and ${{\text{P}}_{\text{2}}}$ have same number of sides. $\Rightarrow {{\text{P}}_{\text{2}}}$ and ${{\text{P}}_{\text{3}}}$ have same number of sides. $\Rightarrow {{\text{P}}_{\text{1}}}$ and ${{\text{P}}_{\text{3}}}$ have same number of sides. $\Rightarrow \left( {{\text{P}}_{\text{1}}}\text{, }{{\text{P}}_{\text{3}}} \right)\in \text{R}$ Therefore $\text{R}$ is tr
Ansitive. Therefore the relation $\text{R}$ is an equivalence relation. The elements in $A$ related to right-angled triangle $\left( \text{T} \right)$ with sides $\text{3, 4}$ and $\text{5}$ are the polygons having $\text{3}$ sides.

14. Let $\text{L}$ be the set of all lines in $\text{XY}$ plane and $\text{R}$ be the relation in $\text{L}$ defined as $\text{R = }\!\!\{\!\!\text{ }\left( {{\text{L}}_{\text{1}}}\text{, }{{\text{L}}_{\text{2}}} \right)\text{: }{{\text{L}}_{\text{1}}}$ is parallel to ${{\text{L}}_{\text{2}}}$$\text{ }\!\!\}\!\!\text{ }$. Show that $\text{R}$ is an equivalence relation. Find the set of all lines related to the line $\text{y=2x+4}$.
Ans: $\text{R = }\!\!\{\!\!\text{ }\left( {{\text{L}}_{\text{1}}}\text{, }{{\text{L}}_{\text{2}}} \right)\text{: }{{\text{L}}_{\text{1}}}$ is parallel to ${{\text{L}}_{\text{2}}}\text{ }\!\!\}\!\!\text{ }$. The relation $\text{R}$ is reflexive as any line ${{\text{L}}_{\text{1}}}$ is parallel to itself, so, $\left( {{\text{L}}_{\text{1}}}\text{, }{{\text{L}}_{\text{1}}} \right)\in \text{R}$. Let $\left( {{\text{L}}_{\text{1}}}\text{, }{{\text{L}}_{\text{2}}} \right)\in \text{R}$. $\Rightarrow {{\text{L}}_{\text{1}}}$ is parallel to ${{\text{L}}_{\text{2}}}$, therefore ${{\text{L}}_{\text{2}}}$ is parallel to ${{\text{L}}_{\text{1}}}$. $\Rightarrow \left( {{\text{L}}_{\text{2}}}\text{, }{{\text{L}}_{\text{1}}} \right)\in \text{R}$ Therefore $\text{R}$ is symmetric. Let $\left( {{\text{L}}_{\text{1}}}\text{, }{{\text{L}}_{\text{2}}} \right)\text{,}\left( {{\text{L}}_{\text{2}}}\text{, }{{\text{L}}_{\text{3}}} \right)\in \text{R}$. $\Rightarrow {{\text{L}}_{\text{1}}}$ is parallel to ${{\text{L}}_{\text{2}}}$ $\Rightarrow {{\text{L}}_{\text{2}}}$ is parallel to ${{\text{L}}_{\text{3}}}$ $\Rightarrow {{\text{L}}_{\text{1}}}$ is parallel to ${{\text{L}}_{\text{3}}}$ $\Rightarrow \left( {{\text{L}}_{\text{1}}}\text{, }{{\text{L}}_{\text{3}}} \right)\in \text{R}$ Therefore $\text{R}$ is tr
Ansitive. Therefore the relation $\text{R}$ is an equivalence relation. Set of all lines related to the line $\text{y=2x+4}$ is the set of all lines that are parallel to the line $\text{y=2x+4}$. Slope of line $\text{y=2x+4}$ is $\text{m = 2}$. Therefore, lines parallel to the given line are of the form $\text{y=2x+c}$, where $\text{c}\in \text{R}$.

15. Let $\text{R}$ be the relation in the set $\left\{ \text{1, 2, 3, 4} \right\}$ given by $\text{R = }\left\{ \left( \text{1, 2} \right)\text{, }\left( \text{2, 2} \right)\text{, }\left( \text{1, 1} \right)\text{, }\left( \text{4, 4} \right)\text{, }\left( \text{1, 3} \right)\text{, }\left( \text{3, 3} \right)\text{, }\left( \text{3, 2} \right) \right\}$. Choose the correct
Answer.

$\text{R}$ is reflexive and symmetric but not tr
Ansitive. $\text{R}$ is reflexive and tr
Ansitive but not symmetric. $\text{R}$ is symmetric and tr
Ansitive but not reflexive. $\text{R}$ is an equivalence relation

Ans: $\text{R = }\left\{ \left( \text{1, 2} \right)\text{, }\left( \text{2, 2} \right)\text{, }\left( \text{1, 1} \right)\text{, }\left( \text{4, 4} \right)\text{, }\left( \text{1, 3} \right)\text{, }\left( \text{3, 3} \right)\text{, }\left( \text{3, 2} \right) \right\}$. Since $\left( \text{a, a} \right)\in \text{R}$, for every $\text{a}\in \left\{ \text{1, 2, 3, 4} \right\}$ The relation $\text{R}$ is reflexive. Since $\left( \text{1, 2} \right)\in \text{R}$ , but $\left( \text{2, 1} \right)\notin \text{R}$ . Therefore $\text{R}$ is not symmetric. $\left( \text{a, b} \right)\text{,}\left( \text{b, c} \right)\in \text{R}\Rightarrow \left( \text{a, c} \right)\in \text{R}$ for all $\text{a, b, c}\in \left\{ \text{1, 2, 3, 4} \right\}$. Therefore $\text{R}$ is tr
Ansitive. Therefore the relation $\text{R}$ is reflexive and tr
Ansitive but not symmetric. The correct
Answer is ($B$) $\text{R}$ is reflexive and tr
Ansitive but not symmetric.

16. Let $\text{R}$ be the relation in the set N given by $\text{R = }\left\{ \left( \text{a, b} \right)\text{: a = b - 2, b > 6} \right\}$ Choose the correct
Answer.

$\left( \text{2, 4} \right)\in \text{R}$ $\left( \text{3, 8} \right)\in \text{R}$ $\left( \text{6, 8} \right)\in \text{R}$ $\left( \text{8, 7} \right)\in \text{R}$

Ans: The given relation is $\text{R = }\left\{ \left( \text{a, b} \right)\text{: a = b - 2, b > 6} \right\}$ Now, Considering $\left( \text{2, 4} \right)\in \text{R}$. Since, $\text{b 6}$, so, $\left( \text{2, 4} \right)\notin \text{R}$. Considering $\left( \text{3, 8} \right)\in \text{R}$. Since $\text{3 }\ne \text{ 8 - 2}$, so $\left( \text{3, 8} \right)\notin \text{R}$. Considering $\left( \text{6, 8} \right)\in \text{R}$. Since $\text{86}$ and $\text{6=8-2}$, so $\left( \text{6, 8} \right)\in \text{R}$. Therefore, the correct
Answer is ($C$)$\left( \text{6, 8} \right)\in \text{R}$.