Relations practice · Mixed difficulty

Option D (4) is correct because there are 4 elements that are present in both A x B and B x A, which are (2,2), (2,3), (3,2), and (3,3).

Option C ($2^{\mathrm{mn}}$) is correct because the power set of a set with mn elements has $2^{\mathrm{mn}}$ elements.

Option A (Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1) is correct because both statements define equivalence relations, but the second statement doesn't explain the first.

Option A (1024) is correct because it is the total number of symmetric relations from A to A, calculated using the formula $2^{\frac{n(n+1)}{2}}$.

The correct answer is 9 because the number of elements in the Cartesian product $A \times B$ is the product of the number of elements in $A$ and $B$.

Option B (Reflexive, transitive but not symmetric) is correct because the relation satisfies the reflexive and transitive properties but fails to be symmetric.

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), making it not symmetric.

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

The correct answer is A ($2y = 91x$) because it correctly calculates the number of one-one functions and establishes the relationship between x and y.

Option B (Reflexive, transitive but not symmetric) is correct because the given relation satisfies the reflexive and transitive properties but fails to be symmetric.

Option C is correct because (3, b) is not an element of A x B since 3 is not in A and b is not in B.

Option B is the correct answer because $R_2$ is not transitive, and therefore not an equivalence relation. An equivalence relation must be reflexive, symmetric, and transitive.

Option C ($R$ is an equivalence relation) is correct because the relation is reflexive, symmetric, and transitive.

Option C is correct because the given relation R is transitive.

Option D (Equivalence) is correct because the relation R satisfies reflexive, symmetric, and transitive properties, making it an equivalence relation.

Option B (Symmetric) is correct because if $|a + b| = a + b$, then $|b + a| = b + a$, implying that if (a, b) is in R, then (b, a) is also in R.

The correct answer is B (10) because it represents the minimum number of ordered pairs required to satisfy the properties of an equivalence relation given the initial conditions.

Option A (Reflexive and symmetric) is correct because |a-a| <= 1 and |a-b| <= 1 implies |b-a| <= 1.

Option A (5) is correct because the number of equivalence relations on a set is equivalent to the number of partitions of the set, which is 5 for a set with 3 elements.

Option A (7) is correct because to make R an equivalence relation, we need to add (1,1), (2,2), (3,3), (2,1), (3,2), (1,3), and (3,1).

Option A (5) is correct because the set X contains five distinct elements: 1, 2, 3, 4, and 5.

Option C is correct because the range of R2, defined by x + 2y = 10, with x and y being natural numbers, is {1, 2, 3, 4}.

The correct answer is C. The relation $R$ is not symmetric because $(2,3) \in R$ but $(3,2) \notin R$.

Option A ($\frac{5}{16}$) is correct because there are 5 relations that are both symmetric and transitive out of a total of 16 possible relations.

Option B ($\{4,3,2,1\}$) is correct because it represents the domain of the inverse relation, which is the range of the original relation.

Option C ($R^{-1} \circ S^{-1}$) is correct because the inverse of the composition of two relations is the composition of their inverses in reverse order.

Option B (Symmetric) is correct because if R = R⁻¹, then for every (a, b) in R, (b, a) is also in R, which is the definition of a symmetric relation.

Option D (None of these) is correct because the relation "a is the father of b" is neither reflexive, symmetric, nor transitive.

The correct answer is B (2¹⁵) because it correctly determines the number of symmetric relations on set A by considering the pairs and their reverses.

Option D (4 is the remainder when α is divided by 7) is correct because it satisfies the conditions for reflexivity, symmetry, and transitivity, making R an equivalence relation.