Transitive Relations

Q.

37. Let A = {1, 2, 3}. Then find the number of equivalence relations containing (1, 2

Q. Given an example of a relation. Which is (i) Symmetric but neither reflexive nor transitive. (ii) Transitive but neither reflexive nor symmetric. (iii) Reflexive and symmetric but not transitive. (iv) Reflexive and transitive but not symmetric. (v) Symmetric and transitive but not reflexive.

Q. Let a set $A=A_1\cup A_2\cup \cdots \cup A_k$, where $A_i\cap Aj=\varphi$ for $i\ne j,1\le i,j\le k$. Define the relation $R$ from $A$ to $A$ by $R=\left\{\right(x,y):y\in A_i\text{if and only if}x\in A_i,1\le i\le k\}$. Then, $R$ is :

Q. Let A be the non – empty set of children in a family. The relation ‘x is a brother of y’ in A is

1. reflexive
2. symmetric
3. transitive
4. an equivalence relation

Q.

Let $R$ be the real line consider the following subsets of the plane$R\times R,S=\left\{\right(x,y):y=x+1\text{and}0<x<2\},T=\left\{\right(x,y):x-y\text{is an integer}}$ .Which one of the following is true?

1. $T$ is an equivalence relation on $R$ but $S$ is not

2. Neither $S$ nor $T$ is an equivalence relation on $R$

3. Both $S$ and $T$ are equivalence relations on $R$

4. $S$ is an equivalence relation on $R$ but $T$ is not

Q.

The domain of the function $f\left(x\right)=\frac{1}{{\log }_{10}(1-x)}+\sqrt{x+2}$ is

1. $]-3,-2.5[\cup ]-2.5,-2[$

2. $[-2,0[\cup ]0,1[$

3. $]0,1[$

4. None of these

Q.

Determine whether each of the following relations are reflexive, symmetric and transitive:
(ii) Relation R in the set N of natural numbers defined as
R = {(x, y): y = x + 5 and x < 4}

Q.

The relation $R=\left\{(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)\right\}$ on set $A=\left\{1,2,3\right\}$ is

1. Reflexive but not symmetric

2. Reflexive but not transitive

3. Symmetric and transitive

4. Neither symmetric nor transitive

Q.

Determine whether each of the following relations are reflexive, symmetric and transitive:
(v) Relation R in the set A of human beings in a town at a particular time given by
(a) R = {(x, y): x and y work at the same place}
(b) R = {(x, y): x and y live in the same locality}
(c) R = {(x, y): x is exactly 7 cm taller than y}
(d) R = {(x, y): x is wife of y}
(e) R = {(x, y): x is father of y}

Q.

Let R and S be two non – void relations on a set A. which of the following statements is false

1. $R,S$ are transitive $\Rightarrow R\cup S$ is transitive

2. $R,S$ are transitive $\Rightarrow R\cap S$ is transitive

3. $R,S$ are symmetric $\Rightarrow R\cup S$ is symmetric

4. $R,S$ are reflexive$\Rightarrow R\cap S$ is reflexive

Q. Explain why every identity relation is equivalent.Give examples

Q. Determine whether each of the following relations are reflexive, symmetric and transitive: (i)Relation R in the set A = {1, 2, 3…13, 14} defined as R = {( x , y ): 3 x − y = 0} (ii) Relation R in the set N of natural numbers defined as R = {( x , y ): y = x + 5 and x < 4} (iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as R = {( x , y ): y is divisible by x } (iv) Relation R in the set Z of all integers defined as R = {( x , y ): x − y is as integer} (v) Relation R in the set A of human beings in a town at a particular time given by (a) R = {( x , y ): x and y work at the same place} (b) R = {( x , y ): x and y live in the same locality} (c) R = {( x , y ): x is exactly 7 cm taller than y } (d) R = {( x , y ): x is wife of y } (e) R = {( x , y ): x is father of y }

Q. If $A=\{A_1,A_2,A_3,A_4,A_5,A_6\}$ be the set of six unit circles with centres at $C_1,C_2,C_3...C_6$ as shown in the diagram. The relation $R$ on $A$ is defined by
$(A_i,A_j)\in R⟺C_i{C}_j\le 2\sqrt{2},i,j\in \{1,\mathrm{2...},6\}$ where $C_i{C}_j$ represents distance between the centres $C_i\text{and}{C}_j$ , then

1. $R$ is symmetric and transitive but not reflexive relation.
2. $R$ is transitive relation only.
3. $R$ is reflexive and symmetric but not transitive relation.
4. $R$ is neither reflexive nor transitive but it is symmetric relation

Q.

Determine whether each of the following relations are reflexive, symmetric and transitive:
(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as
R = {(x, y): y is divisible by x}

Q. Determine whether the following relation is reflexive, transitive and symmetric :
R = {(x, y) : x and y work at the same place }

Q. If $A=\left[\begin{array}{ccc}2& -1& 3\\ -4& 5& 1\end{array}\right]\mathrm{and}B=\left[\begin{array}{cc}2& 3\\ 4& -2\\ 1& 5\end{array}\right]$, then

(a) only AB is defined
(b) only BA is defined
(c) AB and BA both are defined
(d) AB and BA both are not defined

Q.

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

(iv) Relation R in the set Z of all integers defined as R = {(x, y): x − y is an integer}

Q. for the set a={1, 2, 3} a relation r is defined as r={(1, 1), (2, 2), (3, 3), (1, 3)}. write the ordered pairs to be added to R to make the smallest equivalence relation.

Q.

What is transitive relations. Give an example

Q.

Which of the following relations are transitive?

1. "Shares birthday with" on set of people

2. "Is friends on Facebook with" on set of people in fb

3. "Is greater than" on a set of numbers

4. "Is elder to" on a set of people

Q. Write the inverse of 5 under multiplication modulo 11 on the set {1, 2, ... , 10}.

Q. Let N denote the set of all natural numbers. Define two binary relations on N as $R_1=\left\{\right(x,y)ϵN\times N:2x+y=10\}$ and $R_2=\left\{\right(x,y)ϵN\times N:x+2y=10\}$. Then.

1. Both $R_1$ and $R_2$ are transitive relations
2. Both $R_1$ and $R_2$ are symmetric relations
3. Range of $R_2$ is $\{1,2,3,4\}$
4. Range of $R_1$ is $\{2,4,8\}$

Q. In the following question, one part of the sentence may have an error. Choose which part of the sentence has an error and choose the option corresponding to it. If the sentence is free from error, choose ''No error'' option.
I pride on (a)/ being able to work (b)/ smoothly under pressure too. (c)/ No error (d)

1. a
2. b
3. c
4. d

Q. 32. let f(x)=x/1+modx belongs to R then f is options 1.one one 2.even 3.decresing 4. many

Q. Let $R_1$ and $R_2$ be two relations defined as follows:
$R_1=\left\{\right(a,b)\in R^2:a^2+b^2\in Q\}$ and $R_2=\left\{\right(a,b)\in R^2:a^2+b^2\notin Q\}.$ Then

1. Neither $R_1$ nor $R_2$ is a transitive relation
2. $R_1$ is transitive but $R_2$ is not a transitive relation
3. $R_1$ and $R_2$ both are transitive relations
4. $R_2$ is transitive but $R_1$ is not a transitive relation

Q. A relation $R$ is defined as $aRb$ if $“a$ is the father of $b”.$ Then $R$ is

1. reflexive
2. symmetric
3. transitive
4. none of these

Q. Show that the relation R in the set R of real numbers, defined as R = {( a , b ): a ≤ b 2 } is neither reflexive nor symmetric nor transitive.

Q.

Let A ={a, b, c} and the relation R be defined on A as follows.
R ={(a, a), (b, c), (a, b)}

Then, write the ordered pairs to be added in R to make R reflexive and transitive.

Q.

Show that the statement “For any real numbers a and b, a² = b² implies that a = b” is not true by giving a counter-example.

Q. Let A be the non – empty set of children in a family. The relation ‘x is a brother of y’ in A is

1. reflexive
2. symmetric
3. transitive
4. an equivalence relation