Relations practice · Mixed difficulty

The correct answer is D (An equivalence relation) because the relation R is reflexive, symmetric, and transitive, satisfying the definition of an equivalence relation.

Option A is correct because the inverse relation ${R^{ - 1}}$ is obtained by swapping the elements in each ordered pair of the relation R.

Option B (Reflexive and symmetric) is correct because the relation satisfies both reflexive and symmetric properties but not transitive.

Option B ($\{(4,2),(3,4),(2,6),(1,8)\}$) is correct because it correctly represents the inverse relation $R^{-1}$ based on the given relation R.

Option A ($(-\infty,-\frac{21}{4}] \cup[0, \infty)$) is correct because by finding the range of the function, we get $y \in(-\infty,-\frac{21}{4}] \cup[0, \infty)$.

Option D (None of these) is correct because none of the given options fully represent the relation R based on the condition $|a^2 - b^2| < 16$.

Option B ($R_1$ is not reflexive, $R_2$ is reflexive) is correct because $R_1$ requires $a^2 + a^2 = 1$ which is not true for all real numbers, while $R_2$ satisfies the reflexive property.

The correct answer is B. $T$ is an equivalence relation because it is reflexive, symmetric, and transitive, while $S$ is not reflexive and thus not an equivalence relation.

Option D (Equivalence) is correct because the relation is reflexive (everyone speaks the same language as themselves), symmetric (if A speaks the same language as B, B speaks the same language as A), and transitive (if A speaks the same language as B and B speaks the same language as C, then A speaks the same language as C).

Option B (90) is correct because there are 10 choices for x and, since x cannot equal y, there are only 9 choices for y, giving a total of 10 * 9 = 90 elements.

Option D (Neither $p$ nor $q$) is correct because a reflexive relation is not necessarily symmetric, and an antisymmetric relation is not necessarily reflexive.

Option C (An equivalence relation) is correct because the relation R = {(1,1), (2,2), (3,3)} on the set {1,2,3} is reflexive, symmetric, and transitive.

Option B (${2^{12}}$) is correct because the number of reflexive relations on a set with four elements is ${2^{{4^2} - 4}} = {2^{12}}$.

The correct answer is D. $R_4$ is not symmetric because $A \subseteq B$ does not necessarily imply $B \subseteq A$.

Option D (Symmetric and transitive) is correct because the relation satisfies the conditions for symmetry and transitivity, but not reflexivity.

Option C ([6],[13]) is correct because it provides the two solutions to the congruence equation within the modulo 14 system.

Option A (54) is correct because it is the number of sets $B_i$ calculated based on the given conditions of union and element frequency.

Option A (Reflexive but not symmetric) is correct because the relation contains (1,1), (2,2), and (3,3), making it reflexive, but it lacks (2,1) for (1,2) and (3,2) for (2,3), making it not symmetric.

Option C (Transitive only) is correct because the relation R = {(b, c)} satisfies the transitive property, but not reflexive or symmetric.

Assertion is correct, Reason is incorrect because R is a subset of A x A, meaning it *is* a relation on A.

Option D (7) is correct because 7 elements need to be added to the relation R to make it symmetric and transitive.

Option B (Symmetric and transitive) is correct because the void relation is symmetric and transitive by definition, as the conditions for symmetry and transitivity are vacuously true.

Assertion is incorrect, Reason is correct because the number of relations from A to B is 2^(n(A) * n(B)), and in this case, it should be 2^(3*2) = 2⁶, not 2⁵.

Option A (Only I is true) is correct because addition, subtraction, and multiplication are indeed binary operations on the set of real numbers, while division is not.

The correct answer is B. The relation $R$ is symmetric only because if the GCD of $a$ and $b$ is 2, then the GCD of $b$ and $a$ is also 2, but it's not reflexive or transitive.

Statement-II is true because the number of elements in the Cartesian product of two sets is the product of the number of elements in each set, regardless of the order of the sets.

Option B (Symmetric) is correct because for every (a, b) in R, (b, a) is also in R. In this case, (1,3) and (3,1) are both in R.

Option B (Symmetric) is correct because if a is born in a different month than b, then b is also born in a different month than a.

Option D (reflexive, transitive and not symmetric) is correct because if n is a factor of m, then the relation is reflexive and transitive, but not symmetric.

Option C (Symmetric but not transitive) is correct because the relation satisfies the symmetric property but fails the transitive property.