Cartesian product of sets

Correct answer: A. 6

Explanation: In the Cartesian product \(A\times B\), each element of \(A\) is paired with every element of \(B\). Therefore, \(n(A\times B)=n(A)\times n(B)=3\times2=6\), so the correct answer is 6. The values 5, 3 and 2 result from incorrect counting. Exam tip: For finite sets, use the formula \(n(A\times B)=n(A)n(B)\).

Correct answer: A. {(a,0), (a,1), (b,0), (b,1), (c,0), (c,1)}

Explanation: The Cartesian product A × B is defined as the set of all ordered pairs (x, y) such that x belongs to A and y belongs to B. The order is essential: the first coordinate must come from A, while the second must come from B. Since A has three elements and B has two, the product must contain 3 × 2 = 6 ordered pairs. Pairing each element of A with both elements of B gives (a,0), (a,1), (b,0), (b,1), (c,0) and (c,1). Thus option A is correct. Option B reverses the order and represents B × A, option C uses pairs from A only, and option D lists only three of the six required pairs.

Correct answer: A. true

Explanation: Since (2\in A) and (3\in B), ((2,3)\in A\times B) is true. Check both positions separately in an ordered pair.

Correct answer: A. (8)

Explanation: (B\cap C={4,6}), so (n(A\times(B\cap C))=4\times2=8). First find the intersection and then multiply.

Correct answer: A. 8

Explanation: The governing concepts are union cardinality and the product-cardinality rule. First form the union, remembering that a set records each distinct element only once: B ∪ C = {3,4} ∪ {4,5,6} = {3,4,5,6}. The common element 4 is therefore counted once, so n(B ∪ C) = 4. Also, n(A) = 2. For finite sets, n(X × Y) = n(X)n(Y), because every choice from X can be paired with every choice from Y. Hence n(A × (B ∪ C)) = 2 × 4 = 8, making option A correct. Option B results from counting the repeated 4 twice, while 6 and 12 do not follow the required multiplication after the union is formed.

Correct answer: A. 4

Explanation: Set difference A − B consists of elements that are in A but not in B. From A = {1, 2, 3, 4, 5}, remove 1, 3 and 5 because they belong to B; this leaves A − B = {2, 4}, which has two elements. The set C = {0, 2} also has two elements. For finite sets, the Cartesian product rule is n(X × Y) = n(X)n(Y). Therefore n((A − B) × C) = 2 × 2 = 4, so option A is correct. The elements themselves may be listed as (2,0), (2,2), (4,0) and (4,2). Options B, C and D result from failing to remove the elements of B or from using an incorrect counting rule.

Correct answer: A. (7)

Explanation: (n(A\times B)=n(A)n(B)), so (35=5n(B)) and (n(B)=7). In such questions, use the formula in reverse too.

Correct answer: A. both have (12) elements

Explanation: Both have (3\times4=12) elements. The number can be the same, but the order of pairs is generally different.

Correct answer: A. (8)

Explanation: (A={1,2,3,4}) and (B={1,2}), so (n(A\times B)=4\times2=8). Convert set-builder form into roster form first.

Correct answer: A. (8)

Explanation: The direct answer is option A, 8 pairs. First list the integers satisfying -2≤x≤1: A={-2,-1,0,1}, so n(A)=4. Set B={0,2}, so n(B)=2. For every one of the 4 choices from A, there are 2 choices from B, and each choice produces one ordered pair. Hence n(A × B)=n(A)n(B)=4×2=8. Explicitly, the pairs are (-2,0),(-2,2),(-1,0),(-1,2),(0,0),(0,2),(1,0),(1,2). Option A is correct. Option B, 6, results from missing one or more integers in A. Option C, 4, counts only the elements of A and ignores the second coordinate. Option D, 10, overcounts the product. Pay attention that the interval includes both endpoints -2 and 1 because the inequalities are ≤.

Correct answer: A. (\varnothing)

Explanation: No first component can be taken from the empty set, so no ordered pair is formed. If either side is empty, the Cartesian product is (\varnothing).

Correct answer: A. (4)

Explanation: (n(A\times A)=n(A)^2=2^2=4). Pairs with equal components such as ((p,p)) and ((q,q)) are also included.

Correct answer: A. (9)

Explanation: The valid pairs are ((1,2),(1,4),(2,4),(3,4),(1,6),(2,6),(3,6),(4,6)), so the count is (8). List carefully to avoid overcounting.

Correct answer: A. 3

Explanation: For the condition x+y=4, check the possible values of x. When x=1, y=3; when x=2, y=2; and when x=3, y=1. All these pairs belong to A\times B. For x=0, y would have to be 4, but 4 is not in B. Thus the ordered pairs are (1,3), (2,2), and (3,1), giving a total of 3 pairs. Exam tip: determine the second coordinate from the condition and then verify that both coordinates belong to their respective sets.

Correct answer: A. 3

Explanation: For each element of \(A\), calculate \(y=2x\). This gives \(y=2\) for \(x=1\), \(y=4\) for \(x=2\), and \(y=8\) for \(x=4\). Thus, the valid pairs are \((1,2),(2,4),(4,8)\), giving a total of 3 pairs. A pair is valid only when \(x\in A\) and the corresponding \(y\in B\); merely counting the elements of either set is incorrect. Exam tip: for a condition of the form \(y=f(x)\), test every \(x\) from the first set and check whether the resulting \(y\) belongs to the second set.

Correct answer: A. (7)

Explanation: The valid pairs are ((2,1),(2,2),(3,1),(3,3),(6,1),(6,2),(6,3)). In divisibility, note which component is the divisor.

Correct answer: A. (3)

Explanation: When the first component (2) is fixed, the second component can be any of the (3) elements of (B). Hence there are (3) pairs.

Correct answer: A. (4)

Explanation: When the second component (6) is fixed, the first component can be any of the (4) elements of (A). So there are (4) pairs.

Correct answer: A. (2^6)

Explanation: (n(A\times B)=3\times2=6), and every relation is a subset of (A\times B). Therefore the total number of relations is (2^6).

Correct answer: A. ({(1,4),(3,5)})

Explanation: The direct answer is option A: {(1,4),(3,5)}. A relation from A to B is any subset of A × B. This means every ordered pair must have its first entry in A and its second entry in B. Here A = {1,2,3} and B = {4,5}. In option A, (1,4) and (3,5) both satisfy this rule, so the set is a valid relation. Option B reverses the order: (4,1) and (5,3) have first entries from B and second entries from A, so it is a subset of B × A, not A × B. Option C contains (1,6), but 6 is not in B. Option D contains 0, which is not in A, so its first pair is invalid; also the second pair uses 2 as a second component, not an element of B. A relation need not contain every element or every possible pair; it only needs to be a subset. Memory cue: for a relation from A to B, read each pair left to right as A then B.

Correct answer: A. (2^6)

Explanation: (n(A\times B)=2\times3=6), so the number of subsets is (2^6). Use the formula (2^n) for counting subsets.

Correct answer: A. 1

Explanation: The governing concepts are Cartesian-product membership and intersection. In A × B, the first coordinate comes from A and the second from B, so A × B = {(1,2),(1,3),(2,2),(2,3)}. In B × A, the first coordinate comes from B and the second from A, giving B × A = {(2,1),(2,2),(3,1),(3,2)}. The intersection contains only ordered pairs that occur in both lists. The sole common pair is (2,2), because 2 belongs to both A and B. Thus the intersection has exactly one element and option A is correct. Pairs such as (1,2) and (2,1) may look related, but they are different ordered pairs because their coordinate order differs. This order distinction is essential.

Correct answer: A. A line segment on \(y=1\) with \(0\le x\le2\)

Explanation: In the Cartesian product \(A\times B\), the first coordinate comes from \(A\) and the second from \(B\). Thus, \(A\times B=\{(x,1):0\le x\le2\}\), so the value of \(y\) is always 1. Therefore, it is the horizontal line segment from \(x=0\) to \(x=2\) on \(y=1\). Option B interchanges the roles of the two coordinates. Exam tip: when the second set is a singleton \(\{c\}\), the product lies on the line \(y=c\).

Correct answer: A. {(-1, y) : 2 ≤ y ≤ 5}

Explanation: By definition, A × B = {(a,b) : a ∈ A and b ∈ B}. The set A contains only -1, so every ordered pair in the product must have -1 as its first coordinate. The interval B = [2,5] contains every real number y satisfying 2 ≤ y ≤ 5, with both endpoints included. Consequently, A × B = {(-1,y) : 2 ≤ y ≤ 5}, which is option A. Geometrically, the product is the vertical line segment x = -1 between y = 2 and y = 5. Option B reverses the coordinate roles, option C incorrectly fixes the first coordinate at 2, and option D includes only one pair rather than all pairs generated by the interval. Thus A is the only complete description.

Correct answer: A. (4)

Explanation: (n(B\times C)=1\times2=2), so (n(A\times(B\times C))=2\times2=4). Count the inner Cartesian product first.