Transitive Relations

Q.

Let $R$ and $S$ be two equivalence relations on a set $A$. Then

1. $R\cup S$ is an equivalence relation on $A$

2. $R\cap S$ is an equivalence relation on $A$

3. $R-S$ is an equivalence relation on $A$

4. None of the above

Q.

Let$A$ be the set of the children in a family. The relation $‘x’$ is a brother of $‘y’$on $A$ is

1. Reflexive

2. Symmetric

3. Transitive

4. None of these

Q.

If a relation $R$ on the set $N$of natural numbers is defined as$(x,y)\iff x^2-4xy+3y^2=0,x,y$ is an element of $N$. Then, the relation $R$ is

1. reflexive

2. symmetric

3. transitive

4. an equivalence relation

Q.

Let $W$ denote the words in the English dictionary. Define the relation $R$ by $R=\{(x,y)\in W\times W|$the words $x$ and $y$ have at least one letter in common}

1. Not reflexive, symmetric, and transitive

2. Reflexive, symmetric, and not transitive

3. Reflexive, symmetric, and transitive

4. Reflexive, not symmetric, and transitive

Q.

Which of the following defined on Z is not an equivalence relation?

1. $(x,y)\in S\iff x\ge y$

2. $(x,y)\in S\iff x=y$

3. $(x,y)\in S\iff x-y$ is a multiple of $3$

4. $(x,y)\in S$ if $\left|x-y\right|$ is even

Q. Let $A=\{a,b,c\}$ and $\left\{\right(a,a),(b,b),(c,c),(b,c),(a,b\left)\right\}$ be a relation on A, then R is:

1. symmetric
2. reflexive
3. anti-symmetric
4. transitive

Q. Prove that the relation R in set of real number R defined as $R=\left\{\right(a,b):a\ge b\}$ is reflexive and transitive but not symmetric.

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.

The relation ‘is less than’ on a set of natural numbers is

1. Only reflexive

2. Only symmetric

3. only transitive

4. an equivalence relation

Q. Let $N$ denote the set of all natural numbers and $R$ be the relation on $NxN$ defined by $(a,b)R(c,d)\iff ad(b+c)=bc(a+d)$, them $R$ is

1. reflexive only
2. symmetric only
3. transitive only
4. equivalence

Q. Consider the non-empty set consisting of children in a family. state giving reasons whether each of the following relations is (i) Symmetric (ii) Transitive
(a) x is a brother of y.
(b) x likes y.

Q. Consider the set $A=\{a,b,c\}$.
Give an example of a relation R and A which is
(i) reflexive and symmetric but not reflexive.
(ii) symmetric and transitive but not reflexive.
(iii) reflexive and transitive but not symmetric.

Q. Let $R$ be the relation on the set $R$, of all real numbers defined by $aRb$ if $|a-b|\le 1$. Then, $R$ is :

1. Transitive only
2. Anti - symmetric only
3. Symmetric only
4. Reflexive and symmetric

Q. Give examples of relations which are transitive but neither reflexive nor symmetric

Q.

The relation "less than” in the set of natural numbers is

1. Only reflexive

2. Only symmetric

3. Equivalence relation

4. Only transitive

Q. Let R be a relation in a set A. If $(a,a)\in R,$ for all a $\in$ A.then R is called a __

1. Antisymmetric relation
2. Symmetric relation
3. Reflexive relation
4. Antireflexive relation

Q. Let R be a relation on the set N of natural numbers defined by $nRm\iff n$ is a factor of m (i.e., n | m). Then R is

1. Reflexive and symmetric
2. Transitive and symmetric
3. Equivalence
4. Reflexive, transitive but not symmetric

Q. (i)$aRb⟺\left|a\right|=\left|b\right|$
it is Reflexive, not symmetric, transitive.

1. True
2. False

Q. The following relation is defined on the set of real numbers. a R b iff $|a-b|>0.$Choose the correct options ?

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