Cartesian product of sets

Correct answer: A. All ordered pairs of (A) and (B)

Explanation: In (A\times B), the first component comes from (A) and the second from (B). In exams, always check the order of the pair.

Correct answer: A. {(1, 3), (2, 3)}

Explanation: The Cartesian product A × B contains all ordered pairs (a, b) such that a belongs to A and b belongs to B. Since A has 1 and 2 while B has only 3, the pairs are (1, 3) and (2, 3). The first coordinate must come from A, so option A is correct. Option B reverses the order, option C lists elements rather than ordered pairs, and option D forms an incorrect pair.

Correct answer: B. 2 युग्म

Explanation: The governing concept is the Cartesian product. For finite sets A and B, every element of A is paired with every element of B, so the number of ordered pairs is n(A×B)=n(A)×n(B). Since A={5}, it has one element and n(A)=1. Since B={7,8}, it has two elements and n(B)=2. Therefore n(A×B)=1×2=2. Listing the product confirms the calculation: A×B={(5,7),(5,8)}. Thus option B is correct. Option A counts only one element of B, while options C and D overcount the possible pairings. The order within each pair matters, and the first coordinate must come from A while the second must come from B.

Correct answer: B. (20)

Explanation: The direct answer is option B: 20. The Cartesian product A×B is the set of all ordered pairs (a,b) where the first element comes from A and the second comes from B. If A has 4 elements, each of those 4 elements can be paired with every one of the 5 elements of B. Therefore the total number of pairs is n(A×B)=n(A)×n(B)=4×5=20. Option B is correct. Option A, 9, is obtained by adding 4 and 5, but Cartesian-product counting uses multiplication, not addition. Option C, 1, does not represent the number of possible choices and has no basis here. Option D, 45, is not the product 4×5 and therefore cannot be the count. The order also matters in a Cartesian product: (a,b) is generally different from (b,a), though the counting formula remains the product of the two set sizes. Memory cue: for ‘each element with every element’, multiply.

Correct answer: A. (\varnothing)

Explanation: There is no element in (A) for the first component, so no ordered pair is formed. Pay special attention to the empty set.

Correct answer: A. Yes, because (2\in A) and (3\in B)

Explanation: In ((2,3)), the first component (2) is in (A) and the second (3) is in (B). Check both positions separately for membership.

Correct answer: A. (p\in A) and (q\in B)

Explanation: In (A\times B), the first component of an ordered pair is from (A) and the second is from (B). This rule solves most membership questions.

Correct answer: B. (A\times B) and (B\times A) are generally not equal

Explanation: Order matters in Cartesian product, so (A\times B) is generally different from (B\times A). Reversed order is a common exam mistake.

Correct answer: B. (A\times B\ne B\times A)

Explanation: (A\times B={(1,1),(1,2)}) and (B\times A={(1,1),(2,1)}). Similar looking numbers do not remove the importance of order.

Correct answer: A. (5)

Explanation: In ((5,7)), the first component is (5) and the second component is (7). In Cartesian product, the first component comes from the first set.

Correct answer: B. 7

Explanation: In an ordered pair \((a,b)\), the first component is \(a\) and the second component is \(b\). Therefore, the second component of \((5,7)\) is 7. Option C reverses the order of the components, while option D represents a set. In exams, pay close attention to the position of each component.

Correct answer: C. (4)

Explanation: (A) has (2) elements and (B) has (2) elements, so (2\times 2=4). For small sets, you may also list all pairs to check.

Correct answer: B. ((3,5))

Explanation: In ((3,5)), (3\in A) and (5\in B), so it belongs to (A\times B). Other options have wrong position or membership.

Correct answer: D. ((3,1))

Explanation: In ((3,1)), the first component (3) is not in (A). To find the wrong option, check the first component first.

Correct answer: A. ({(0,1),(0,2),(0,3)})

Explanation: The only element (0) of (A) stays in the first position and all elements of (B) come in the second position. Do not reverse order even for singleton sets.

Correct answer: C. 9

Explanation: For a Cartesian product, \(n(A\times B)=n(A)\times n(B)\). Since both sets here are \(A\), \(n(A\times A)=3\times3=9\). Option 6 is incorrect because the cardinalities are multiplied, not added. Exam tip: the Cartesian product of an \(n\)-element set with itself contains \(n^2\) ordered pairs.

Correct answer: B. 4

Explanation: For two finite sets, n(A × B) = n(A) × n(B). Thus, 12 = 3 × n(B), so n(B) = 12 ÷ 3 = 4. Option 9 is incorrect because the unknown cardinality is found by dividing the product by the known cardinality, not by multiplying them. Exam tip: The number of ordered pairs in a Cartesian product equals the product of the cardinalities of the two sets.

Correct answer: A. (B=\varnothing)

Explanation: If (A) is not empty and still no pair is formed, then (B) must be empty. If a Cartesian product is empty, at least one set is empty.

Correct answer: A. All pairs with first component from (A) and second component from (B)

Explanation: In (A\times B), the first position is filled from (A) and the second from (B). Confusing it with (A\cup B) is a common mistake.

Correct answer: B. ((1,2)\ne (2,1))

Explanation: Changing positions changes an ordered pair. Equality holds only when corresponding components are equal.

Correct answer: A. ({(1,1),(1,2),(2,1),(2,2)})

Explanation: In (A\times A), every element of (A) pairs with every element of (A). Taking only equal component pairs is incomplete.

Correct answer: A. (0)

Explanation: There is no element in (B), so no second component is available. Since the number is asked, the answer is (0).

Correct answer: B. (1)

Explanation: Both sets have one element each, so only ((4,6)) is formed. The product of singleton sets gives one ordered pair.

Correct answer: B. (a=2,\ b=3)

Explanation: The direct answer is B: a=2 and b=3. Ordered pairs are equal only when their corresponding components are equal. Thus, from (a,3)=(2,b), compare first components: a=2. Compare second components: 3=b, so b=3. Option B is correct. Option A reverses the positions and treats the pair as unordered; an ordered pair is not like an ordinary set. Option C gives the wrong second component. Option D gives the wrong first component. The comma order matters: first coordinate matches first coordinate, and second matches second. This rule is also used in Cartesian products and coordinate geometry.

Correct answer: A. ({(x,y):x\in A,\ y\in B})

Explanation: By definition, (A\times B={(x,y):x\in A,\ y\in B}). In set-builder form, keep the order of components unchanged.