Transitive Relations

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. $R_1$ and $R_2$ both are transitive relations
3. $R_1$ is transitive but $R_2$ is not a transitive relation
4. $R_2$ is transitive but $R_1$ is not a transitive relation

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

1. Reflexive and Symmetric relation only
2. Reflexive but not Transitive relation
3. Equivalence relation
4. Reflexive relation only

Q.

Let $f\left(x\right)$ be a non-constant twice differentiable function defined on $\left(-\infty ,\infty \right)$ such that $f\left(x\right)=f(1-x)$ and$f\left(\frac{1}{4}\right)=0$, Then,

1. $f\left(x\right)$ vanishes at least twice $[0,1]$

2. $f\left(\frac{1}{2}\right)=0$

3. ${\int }_{-0.5}^{0.5}f\left(x+\frac{1}{2}\right)\sin xdx=0$

4. ${\int }_0^{0.5}f\left(t\right)e^{\mathrm{sin\pi }t}dt={\int }_{0.5}^{1}f(1-t)e^{\mathrm{sin\pi }t}dt$

Q. With reference to a universal set, the inclusion of a subset in another, is relation, which is

1. Symmetric only
2. Equivalence relation
3. Reflexive only
4. None of these

Q. Consider the following two binary relations on the set $A=\{a,b,c\}:$
$R_1=\left\{\right(c,a),(b,b),(a,c),(c,c),(b,c),(a,a\left)\right\}\text{and}{R}_2=\left\{\right(a,b),(b,a),(c,c),(c,a),(a,a),(b,b),(a,c\left)\right\}.$
Then :

1. $R_2$ is symmetric but it is not transitive.
2. both $R_1$ and $R_2$ are transitive.
3. both $R_1$ and $R_2$ are not symmetric.
4. $R_1$ is not symmetric but it is transitive.

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. Let $A=\{1,2,3\}$ and $R,S$ be two relations on $A$ given by $R=\left\{\right(1,1),(2,2),(3,3),(1,2),(2,1\left)\right\},S=\left\{\right(1,1),(2,2),(3,3),(2,3),(3,2\left)\right\}$ then $R\cup S$ is

1. Reflexive, symmetric and transitive relation
2. reflexive and transitive relation only
3. not a transitive relation
4. Reflexive relation but not Symmetric relation

Q.

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

1. R and S are transitive then R $\cup$ S is not transitive

2. R and S are transitive then R $\cap$ S is transitive

3. R and S are symmetric then R $\cup$ S is symmetric

4. R and S are reflexive then R $\cup$ S is reflexive

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

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

Q. Consider the set $A=\{1,2,3\}$ and the relation on $A$ as $R=\left\{\right(1,2),(1,3\left)\right\},$ then $R$ is

1. a reflexive relation
2. a symmetric relation
3. a transitive relation
4. None of the above

Q. Let $R$ be a relation defined on $N$ as $R=\left\{\right(x,y):x,y\in N,2x+y=41\}.$ Then $R$ is

1. a symmetric relation
2. both symmetric and transitive relation
3. Neither reflexive nor symmetric nor transitive relation
4. Symmetric relation but not transitive relation

Q.

Let R be a relation on the set N be defined by $\left\{\right(x,y)|x,y|N,2x+y=41\}$. Then R is

1. Reflexive

2. Symmetric

3. Transitive

4. None of these

Q. Which of the following is not correct for relation $R$ on the set of real numbers?

1. $(x,y)\in R\iff |x|–|y|\le 1$ is reflexive but not symmetric
2. $(x,y)\in R\iff 0<|x|–|y|\le 1$ is neither transitive nor symmetric
3. $(x,y)\in R\iff 0<|x–y|\le 1$ is symmetric and transitive
4. $(x,y)\in R\iff |x–y|\le 1$ is reflexive and symmetric

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