Relations practice · Mixed difficulty

Option B is correct because the relation R is defined as a differs from b by 3, meaning a - b = 3, which is satisfied by the elements in option B.

The correct answer is A ({(a, c), (a, d)}) because it is the Cartesian product of (A - B) and (B ∩ C).

Option A (15) is correct because solving the equation $2^m - 2^n = 56$ leads to $m=6$ and $n=3$, so $2m+n = 15$.

Option B (Reflexive) is correct because for any x, x² = x*x, so the relation holds for all elements related to themselves.

Option B is correct. By considering the cases where x >= 9 and 0 <= x < 9, we find that the equation has two solutions: x = 4 and x = 16.

Option A (Reflexive but not symmetric) is correct because a number is divisible by itself, but if b is divisible by a, a is not necessarily divisible by b.

Option B ($\left\{(8,11),(10,13)\right\}$) is correct because it represents the inverse relation $R^{-1}$, where the ordered pairs are reversed from the original relation $R$.

Option B (16) is correct because the relation R has 4 elements, and the number of elements in the power set of a set with n elements is $2^n$, so $2^4 = 16$.

Option A ($a R_1 b \Leftrightarrow |a|=|b|$) is correct because it satisfies the reflexive, symmetric, and transitive properties required for an equivalence relation.

Option A (219) is correct because the number of subsets with at least three elements is found by subtracting the number of subsets with zero, one, and two elements from the total number of subsets.

Option C (Symmetric) is correct because the relation R satisfies the condition for symmetry: if (a, b) is in R, then (b, a) must also be in R.

Option A (an equivalence relation) is correct because the relation R satisfies the reflexive, symmetric, and transitive properties.

Option C ($\left\{ {\left( {1,\;a} \right),\;\left( {1,\;b} \right),\;\left( {2,\;a} \right),\;\left( {2,\;b} \right),\;\left( {3,\;a} \right),\;\left( {3,\;b} \right)} \right\}$) is correct because it represents the Cartesian product of A and B, where each element of A is paired with each element of B.

Option B ($A=\{a, b\}$ and $B=\{1,2,3\}$) is correct because A consists of all the first elements of the ordered pairs in $A \times B$, and B consists of all the second elements.

Option D (${R_4} = \{ (1,3),(2,5),(2,4),(7,9) \}$) is correct because it contains an element (7,9) where 7 is not in X, violating the definition of a relation from X to Y.

Option A is correct because $T$ satisfies reflexive, symmetric, and transitive properties, making it an equivalence relation, while $S$ is not symmetric.

Option B (Symmetric) is correct because if (x, y) is in R, then x² + y² = 1, which implies y² + x² = 1, so (y, x) is also in R.

The correct answer is B (Equivalence relation) because the relation R satisfies reflexive, symmetric, and transitive properties.

The correct answer is C ({(3,3),(3,5),(5,3),(5,5)}) because it represents the composition of the relation R with its inverse.

Option B (9) is correct because it is the square of the number of elements in the intersection of A and B.

Option B ($99^2$) is correct because the number of common elements in $A \times B$ and $B \times A$ is the square of the number of common elements in A and B.

Option D (36) is correct because the maximum number of elements in any relation on A is the number of elements in $A \times A$, which is $6 \times 6 = 36$.

Option B (Reflexive and transitive) is correct because every integer is a multiple of itself and if m is a multiple of n and n is a multiple of p, then m is a multiple of p.

Option D (None of these) is correct because the inclusion of a subset in another is reflexive and transitive, but not symmetric.

The correct answer is B. The relation $R$ is reflexive and symmetric but not transitive because having a common letter between two pairs of words doesn't guarantee a common letter between the first and last word.

Both I and II are true because a relation is not a function if one element in the domain maps to multiple elements in the range, and it is a function if each element in the domain maps to a unique element in the range.

Option C ($\mathrm{A} \times \mathrm{B}$) is correct because a relation from set A to set B is a subset of the Cartesian product A × B.

Option A (0) is correct because substituting the given values into the function allows us to solve for b and c, and then find f(1) = 0.

Option A (5) is correct because to make R an equivalence relation, we need to add (1,1), (3,3), (5,5) for reflexivity, (3,1) for symmetry, and (1,5), (5,3), (5,1) for transitivity, resulting in adding 5 new elements to the original relation.

Option C ($A=\{1,2,3\}$ and $B=\{4,6\}$) is correct because it satisfies the condition that $A \times B$ has 6 elements and includes the given elements.