Relations practice · Easy

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 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 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 (9) is correct because it is the square of the number of elements in the intersection of 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.

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.

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 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.

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 (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.