Relations practice · Mixed difficulty

Option D (None of the above) is correct because the provided options do not fully represent the relation $R$ defined by $|x^2 - y^2| < 16$ on the set $A = \{1, 2, 3, 4, 5\}$.

Option C ($R_3=\{(p, 1),(q, 2),(r, 2),(r, 2)\}$) is not a function because the element 'r' in A is mapped to two different elements in B.

Option C ($\{(1,2),(5,1),(3,1)\}$) is correct because the inverse relation $R^{-1}$ is obtained by swapping the elements in each ordered pair of $R$.

Option A ($R=\{ (1,3),(1,5),(2,3),(2,5),(3,5) \}$) is correct because it lists all pairs (x, y) where x is from A, y is from B, and x < y.

Option D is correct because the subset relation ⊆ on P(A) is reflexive, antisymmetric, and transitive.

Option C ($\\{9,6,3\\}$) is correct because substituting $y = 1, 2, 3$ into the equation $x + 3y = 12$ yields $x = 9, 6, 3$ respectively, and these are the only natural number solutions.

Option A ($2^9$) is correct because each element in $A \times A$ can either be in the relation or not, leading to $2^9$ possible relations.

Option D (Neither $p$ nor $q$) is correct because the identity relation on a set A is both reflexive and symmetric.

The correct answer is C (Not symmetric but reflexive) because the relation satisfies the reflexive property but not the symmetric property.

Option C ($n^2$) is correct because the number of elements common to $A \times B$ and $B \times A$ is the square of the number of elements common to $A$ and $B$.

Option B (Symmetric) is correct because if line α is perpendicular to line β, then line β is also perpendicular to line α.

The correct answer is B (17) because the number of elements in A must be a factor of 45, and 17 is not a factor of 45.

Option D ($i R 1$) is correct because the absolute value of $i$ is equal to 1, satisfying the given relation.

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

Option D ($\{\phi\}$) is correct because the function is not one-to-one, and thus does not have a well-defined inverse at x=2.

Option C is correct because the composition of the relation R with its inverse R^-1 results in the set {{(3,3), (3,5), (5,3), (5,5)}} based on the given conditions.

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 C ($A=\{-1,0,1\}$) is correct because it is the only set that contains -1, 0, and 1, and has 3 elements, resulting in $A \times A$ having 9 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 C ({ (6,0)(0,6) }) is correct because it contains the pairs of non-negative integers that satisfy the equation x² + y² = 36.

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 A (36) is correct because the number of elements in the Cartesian product A x A x B is the product of the number of elements in each set.

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.