Symmetric Relations

Q.

Let $S$ be the set of all real numbers. Then what is the relation $R=\left\{\right(a,b):1+ab>0\}$ on $S$.

1. Reflexive and symmetric but not transitive

2. Reflexive and transitive but not symmetric

3. Symmetric, transitive but not reflexive

4. Reflexive, transitive and symmetric

5. None of the above is true

Q. Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

Q.

Let $R$ be a relation over the set $N\times N$ and it is defined by $(a,b)R(c,d)\Rightarrow a+d=b+c.$ Then $R$ is

1. Reflexive only

2. Symmetric only

3. Transitive only

4. An equivalence relation

Q.

Show that the relation R in the set R of real numbers, defined as $R=\left\{\right(a,b):a\le b^2\}$ is neither reflexive nor symmetric nor transitive.

Here, the result is disproved using some speicific examples. In order to prove a result. we must prove it in generlity and in order to disprove a result we can just provide one example. where the condition is false. It is important to pick up the examles suitably. Since there are certain ordered pairs like (1, 1) for which the relation is reflexive.

Q. Show that the relation R in R defined as R = {( a , b ): a ≤ b }, is reflexive and transitive but not symmetric.

Q.

Let $R$ be a relation on set $A$ such that $R=R^{-1}$, then $R$ is

1. Reflexive

2. Symmetric

3. Transitive

4. None of these

Q.

Show that the relation R in the set A of all the books in a library of a college, given by R = {(x, y): x and y have same number of pages} is an equivalence relation.

Q. Let $A=\{2,3,4,5\}$ be a set and $R=\left\{\right(2,2),(3,3),(4,4),(5,5),(2,3),(3,2),(5,3),(3,5\left)\right\}$ be a relation on $A$. Then $R$ is:

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

Q. Mark the correct alternative in the following question:

Consider a non-empty set consisting of children in a family and a relation R defined as aRb if a is brother of b. Then, R is

(a) symmetric but not transitive (b) transitive but not symmetric
(c) neither symmetric nor transitive (d) both symmetric and transitive

Q.

Let $R$ be a relation of the set of integers given by $aRb$ $\Rightarrow a=2^k.b$ for some integers $k$. Then $R$ is

1. An equivalence relation

2. Reflexive but not symmetric

3. Reflexive and transitive but not symmetric

4. Reflexive and symmetric but not transitive

Q.

Which of the following statements is not correct for the relation $R$ defined by $aRb$, if and only if $b$ lives within one kilometre from $a$?

1. $R$ is reflexive

2. $R$ is symmetric

3. $R$ is not anti-symmetric

4. None of the above

Q. What is the difference between symmetric and reflexive relation

Q.

Which of the below statements is/are correct?

1. Empty relation is a symmetric relation

2. "is married to" is an example of symmetric relation

3. A relation R is defined as ‘x is a divisor of y’

4. Empty relation is a reflexive relation

Q.

The relation {(a, a), (b, b), (c, c)} on the set {a, b, c} is a -

1. Reflexive Relation

2. Symmetric Relation

3. Transitive Relation

4. Equivalence Relation

5. all the above

Q. What is the general formula for number of symmetric relations of a set with n elements?

Q.

Statement I: the curve $y=-\frac{x^2}{2}+x+1$ is symmetric with respect to the line $x=1$.

Statement II: A parabola is symmetric about its axis.

1. Statement I is correct, Statement II is correct; Statement II is a correct explanation for Statement I

2. Statement I is correct, Statement II is correct; Statement II is not correct explanation for Statement I

3. Statement I is correct, Statement II is incorrect

4. Statement I is incorrect, Statement II is correct

Q.

Let $\mathrm{W}$ denote the words in the English dictionary. Define the relation $\mathrm{R}$ by $R=\left\{\left(x,y\right)\in W\times W\mathrm{the}\mathrm{words}\mathrm{x},\mathrm{y}\mathrm{have}\mathrm{atleast}\mathrm{one}\mathrm{letter}\mathrm{in}\mathrm{common}\right\}$ then $\mathrm{R}$ is.

1. reflexive, symmetric, and not transitive

2. reflexive, symmetric, and transitive

3. reflexive, not symmetric, and transitive

4. not reflexive, symmetric, and transitive

Q.

Write the smallest equivalence relation on the set $\{1,2,3\}$.

Q.

If $x$ varies directly as $y$ and $x=12$ when $y$ is $48$ then$x=6$ when $y$ is?

Q.

Let A = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is

(A) 1 (B) 2 (C) 3 (D) 4

Q. Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {( a , b ): b = a + 1} is reflexive, symmetric or transitive.

Q. The relation R = {(1, 1), (2, 2), (3, 3)} on the set {1, 2, 3} is
(a) symmetric only
(b) reflexive only
(c) an equivalence relation
(d) transitive only

Q. Mark the correct alternative in the following question:

Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as aRb if a is congruent to b for all a, b $\in$ T. Then, R is

a) reflexive but not symmetric (b) transitive but not symmetric
c) equivalence (d) none of these

Q. Check whether the relation R on R defined by R = {(a, b) : a ≤ b³} is reflexive, symmetric or transitive.

Q. Symmetric and transitive but not reflexive.

Q. Show that the relation $R$ on $R$ defined as $R=\left\{\right(a,b):a\le b\},$ is reflexive, and transitive but not symmetric.

Q.

Universal relation defined on a set A is Symmetric

1. True

2. False

Q. Is it true that every relation which is symmetric and transitive is also reflexive? Give reasons.

Q. S is a relation over the set R of all real numbers and it is given by
(a, b) ∈ S ⇔ ab ≥ 0. Then, S is
(a) symmetric and transitive only
(b) reflexive and symmetric only
(c) antisymmetric relation
(d) an equivalence relation

Q. Check whether the relation R in R defined as R = {( a , b ): a ≤ b 3 } is reflexive, symmetric or transitive.