Relations practice · Mixed difficulty

Option C ($h=\{\left(n, \frac{{1}}{{n}}\right) \mid n \in \mathrm{{I}}\}$) is correct because when n=0, 1/n is undefined, thus violating the function definition.

Option A (${2^{15}} - {2^{10}}$) is correct because it represents the total number of symmetric relations minus the number of reflexive relations on the set A.

Option B is correct because the union of two transitive relations is not necessarily transitive, as demonstrated by the given counterexample.

Option C (Transitive) is correct because if x is a brother of y, and y is a brother of z, then x is a brother of z.

Option B (660) is correct because there are 11 possible values for b (product of two primes >= 3) and 60 possible values for a, resulting in 60*11 = 660 elements.

Option C (Transitive) is correct because for the given relation R, if (a, b) and (b, c) are in R, then (a, c) is also in R. In this case, the only relevant pair is (1, 2), and there is no (2, x) in R other than (2, 2), and (1, 2) is already in R.

Option C (II and III) is correct because a set lists distinct elements without repetition, and the order of elements does not matter.

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

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 is correct because the relation is symmetric (if x+y=7, then y+x=7), but it's neither reflexive (e.g., 1+1≠7) nor transitive (e.g., (1,6) and (6,1) are in R, but (1,1) is not).

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 because the relation R is symmetric but neither reflexive nor transitive, as demonstrated by the provided counterexamples.

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. $S$ is an equivalence relation because it is reflexive, symmetric, and transitive.