Relations practice · Easy

Option D ($\phi$) is correct because the intersection of the two Cartesian products is an empty set since A and B have no common elements.

Option D (An equivalence relation) is correct because the relation R is reflexive, symmetric, and transitive, satisfying the conditions for an equivalence relation.

Option D (None of these) is correct because the relation R is neither reflexive, symmetric, nor transitive, as demonstrated by the counterexamples provided.

Option D (An equivalence relation) is correct because the relation R is reflexive, symmetric, and transitive.

Option A (Reflexive but neither symmetric nor transitive) is correct because the relation satisfies the reflexive property but fails to satisfy symmetric and transitive properties.

Option D (Equivalence) is correct because the relation is reflexive (an ant walks on the same line as itself), symmetric (if ant A walks on the same line as ant B, then ant B walks on the same line as ant A), and transitive (if ant A walks on the same line as ant B, and ant B walks on the same line as ant C, then ant A walks on the same line as ant C).

Option D (4) is correct because the expression is an integer when n is a factor of 2, including both positive and negative factors.

Option D (Neither $R_1$ nor $R_2$ is an equivalence relation) is correct because neither relation satisfies all three properties of an equivalence relation: reflexivity, symmetry, and transitivity.

Option D (None of these) is correct because none of the given relations satisfy the condition $|x^2 - y^2| < 16$ for all pairs.

Option A ($m \geqslant n$) is correct because a reflexive relation on a set with n elements must have at least n ordered pairs.

Option A (0) is correct because the number of elements in an empty set, denoted by $\phi$, is zero.

Option B is correct because the relation R is reflexive (contains (2,2), (3,3), (4,4), (5,5)) and symmetric (if (a,b) is in R, then (b,a) is also in R).

All three statements are true because they correctly describe the number of relations, the Cartesian product of union and intersection, and the equality of ordered pairs.

Option A is correct because the relation R is reflexive, symmetric, and transitive, thus satisfying the conditions for an equivalence relation.

Option A (2) is correct because there are only two equivalence relations that satisfy the given conditions: the smallest one containing (1,2) and (2,1), and the universal relation.

Option D is correct because it accurately represents the given relation in set-builder form, defining the relationship between elements and their domain.

Option A (1) is correct because there is only one relation that contains (1,2) and (1,3), which is reflexive and symmetric but not transitive.

Option B (Symmetric only) is correct because if a is not equal to b, then b is not equal to a, but a is not equal to itself is false, and transitivity doesn't hold.

7 is the correct answer because the number of relations on X x Y is 2^(n(X) * n(Y)), so 5m = 5 * 7, thus m = 7.

Option A ($2^{35}$) is correct because the ratio of reflexive relations to symmetric relations on a set with 10 elements is calculated as 2^(n2-n) / 2^((n²⁺n)/2) = 2^((n^2-3n)/2), and substituting n=10 gives 2³⁵.

Option B ($I \subset R$) is correct because since R is reflexive, it must contain all the pairs in the identity relation I, meaning I is a subset of R.

Option A ($2^8$) is correct because the number of relations from set A to set B is $2^{n(A) \times n(B)}$, where n(A) and n(B) are the number of elements in sets A and B, respectively.

Option C ($\{0, \pm 3, \pm 4, \pm 5\}$) is correct because it includes all possible integer values of 'a' for which there exists an integer 'b' such that $a^2 + b^2 = 25$.

Option C is correct because the relation R is transitive, meaning if x < y and y < z, then x < z.

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

Option D (Reflexive and transitive only) is correct because the relation R satisfies the reflexive and transitive properties but not the symmetric property.

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

Option A (Inside the rectangle having vertices at $(1,1),(0,1),(0,-1)$ and $(1,-1)$) is correct because it represents the region defined by the Cartesian product of the given intervals for A and B.

Option A (36) is correct because the percentage of people who like both coffee and tea, x, must be greater than or equal to 38 and less than or equal to 65, thus 36 is not a possible value.

Option C (Reflexive and transitive but not symmetric) is correct because the relation satisfies the reflexive and transitive properties but fails the symmetric property.