Relations practice · Easy

9 is the correct answer because the total number of distinct relations that can be defined over X is 2^(n(X x X)), and since n(X) = 3, n(X x X) = 3 * 3 = 9.

Option D (Range of R={5}) is false because the range of R is {1, 2, 3}, not {5}, as determined by the elements present in the relation R.

Option B (Symmetric) is correct because if $(x, y)$ is in $P$, then $x^2 + y^2 = 1$, which implies $y^2 + x^2 = 1$, so $(y, x)$ is also in $P$.

Option C (Symmetric relation) is correct because if a - b is irrational, then b - a is also irrational, satisfying the symmetric property.

Option B (Symmetric) is correct because if $(x, y)$ belongs to $P$, then $x^2 + y^2 = 1$, which implies $y^2 + x^2 = 1$, so $(y, x)$ also belongs to $P$.

Option B (Symmetric) is correct because if (a, b) is in R, then a + b = 4, which means b + a = 4, so (b, a) is also in R.

Option B ($R_2=\{(a, 1),(d, 3),(b, 2),(b, 3)\}$) is correct because all ordered pairs in $R_2$ have their first element from A and their second element from B, thus satisfying the definition of a relation from A to B.

Option C (Equivalence) is correct because the inverse of an equivalence relation is also an equivalence relation.

Option B ($n(A-B)=n(B-A)$) is correct because if A and B have the same number of elements, then the number of elements in A that are not in B is equal to the number of elements in B that are not in A.

Option B (24) is correct because the number of bijections from A to B when n(A) = n(B) = 4 is 4! = 24.

Option B (Symmetric) is correct because if (x, y) is in P, then (y, x) is also in P, since $x^2 + y^2 = y^2 + x^2 = 1$.

Option B (7) is correct because to make the relation an equivalence relation, we need to add 7 elements to ensure reflexivity, symmetry, and transitivity.

Option D ($\phi$) is correct because the intersection of Y x A and Y x B is Y x (A ∩ B), and since A and B are disjoint, their intersection is the empty set.

Option A (Reflexive and symmetric) is correct because the relation R contains all pairs (a, a) for a in A, and for every (a, b) in R, (b, a) is also in R.

Option C is correct. The relation R is reflexive because (x,x) satisfies the equation, but it's neither symmetric nor transitive as demonstrated by the given examples.

Option C (6) is correct because the number of ordered pairs in the equivalence class is determined by finding pairs (a, b) such that 2a = 3b, where a and b are elements of A.

Option B (Symmetric) is correct because if a is a factor of b, it does not necessarily mean that b is a factor of a.

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

Statement I is true, Statement II is false. The number of reflexive relations on a set A with n elements is 2^{(n2 - n)}.

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 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.