Cartesian product of sets

In (A\times B), the first component comes from (A) and the second from (B). While listing, pair each first-set element with every element of (B).

The number of ordered pairs in a Cartesian product is given by n(P×Q)=n(P)×n(Q). Here, n(P)=3 and n(Q)=1, so n(P×Q)=3×1=3. In fact, P×Q={(0,1),(4,1),(8,1)}. Option A gives only the number of elements in Q, while options C and D result from incorrect counting. Exam tip: multiply the cardinalities of the two sets to find the cardinality of their Cartesian product.

The governing definition is A × B = {(a, b) : a belongs to A and b belongs to B}. The order matters: the first component must come from A and the second component must come from B. Since A contains only 7, every ordered pair must begin with 7. Pairing 7 with each element of B gives (7, 2), (7, 4), and (7, 6), so option A is correct. Option B reverses the components and represents B × A. Option C is merely a collection of individual elements, not a set of ordered pairs. Option D includes pairs whose components do not follow the membership rule. Thus the definition and the order of the factors uniquely determine option A.

In ((3,1)), the first component is (3), but (3\notin X). Check the first and second positions separately for membership.

The direct answer is option A: the first component u must come from set A. A Cartesian product A × B is the set of all ordered pairs (a,b) where a belongs to A and b belongs to B. Therefore, if (u,v) belongs to A × B, its position tells us that u is selected from the first set and v from the second set. Option A is correct because u ∈ A. Option B is wrong because B supplies the second component, not generally the first. Option C is wrong because u need not belong to both sets; A and B may have no common element. Option D is wrong because belonging to A ∪ B is not the defining condition for the first coordinate; u must specifically be an element of A. Remember: in an ordered pair from A × B, first with first and second with second.

There is no element in (S) for the second component, so no pair is formed. Product with an empty set gives the empty set.

For two finite sets, the number of ordered pairs in their Cartesian product is given by \(n(A\times B)=n(A)\times n(B)\). Thus, \(n(A\times B)=6\times2=12\), so option B is correct. Remember that the cardinalities are multiplied, not added, for a Cartesian product.

In (B\times A), the first component is (c) from (B), and the second is (a) or (b) from (A). Reversed order gives a different answer.

By definition, (A\times B={(x,y):x\in A,\ y\in B}). Do not change the order of components in set-builder form.

Two ordered pairs are equal only when their corresponding components are equal. Comparing the first components gives \(m=4\), and comparing the second components gives \(8=n\), so \(n=8\). Option A incorrectly reverses the component order. Exam tip: The order of elements matters in an ordered pair.

The governing cardinality rule for a Cartesian product is n(A × B) = n(A) × n(B). Set A has two elements, 10 and 20, so there are two choices for the first component. Set B has three elements, 30, 40, and 50, so there are three choices for the second component. Every first-component choice can be paired with every second-component choice, giving 2 × 3 = 6 ordered pairs. Therefore, option B is correct. The value 5 is not produced by the product rule, 10 is the sum of the two cardinalities rather than their product, and 20 has no valid counting basis. The order of each pair does not change the total count here, but it does determine the actual members of A × B.

A set is defined by its elements, not by the order in which those elements are written. Both A and B contain exactly the elements 1 and 4, so A = B even though their roster forms list the elements in reverse order. Once A and B are equal, replacing B by A in a Cartesian product gives A × B = A × A. This product is not empty; it contains (1,1), (1,4), (4,1), and (4,4). Therefore, option A is correct. Option B incorrectly treats a set like an ordered list, while options C and D wrongly claim that the products are empty despite both sets being non-empty.

In ((4,3)), (4\in A) and (3\in B), so it is correct. Having the elements present is not enough; their positions must be correct.

(15=5\times n(B)), so (n(B)=3). To find the unknown count, divide the total pairs by the known count.

There is no element in (A) for the first component, so no pair is formed. Since the number is asked, the answer is (0).

Both components of ((2,1)) are from (A), so it is in (A\times A). In (A\times A), both positions use elements of (A).

The direct answer is option C, 16. The set A = {1,2,3,4} has four elements, so n(A)=4. In A × A, the first entry can be chosen in 4 ways and, independently, the second entry can also be chosen in 4 ways. By the product rule, n(A × A)=n(A)n(A)=4×4=16. Option A, 4, counts only one position and misses the choices for the other position. Option B, 8, has no valid counting basis and is not 4×2 because both positions use A. Option C is correct because every one of the 16 ordered pairs is included, such as (1,1), (1,2), and so on. Option D, 20, overcounts the possible pairs. The order matters: (1,2) and (2,1) are different pairs. Memory cue: for a set of n elements, A × A contains n² pairs.

If (B) is not empty and the product is still empty, then (A) must be empty. If a Cartesian product is empty, at least one set is empty.

The governing concept is the definition of a Cartesian product and the order of an ordered pair. For sets P and Q, (x, y) belongs to P × Q exactly when x belongs to P and y belongs to Q. Here P = {6}, a singleton set, so the first component x must be 6. The complete Cartesian product is {(6, 2), (6, 3), (6, 4)}. Thus every possible ordered pair has x = 6, while y may be 2, 3, or 4. Therefore option C is correct. Options A, B, and D are possible values of the second component y, not the first component x. Confusing the order of the sets or reading the pair backwards leads to those distractors. The notation P × Q must always be read from left to right.

The direct answer is option C, y=5. In the Cartesian product {2,4} × {5}, the first coordinate must be selected from {2,4}, while the second coordinate must be selected from {5}. Thus the only possible ordered pairs are (2,5) and (4,5). In both pairs the second coordinate is 5, so y=5. Option A, 2, is wrong because 2 can occur only as the first coordinate here. Option B, 4, is wrong for the same reason: 4 belongs to the first set and cannot be the second coordinate. Option C is correct because the second set contains only 5. Option D, 2 or 4, confuses the first-coordinate choices with the second-coordinate choice. Read the product from left to right: the first set controls x and the second set controls y.

The first component (3) is fixed and the second component is chosen from the (2) elements of (B). The number of such pairs is (n(B)).

The second component (3) is fixed and the first component can be any of the (2) elements of (A). In such questions, keep the fixed component separate.

Both have (2\times 2=4) pairs, but the order of components differs. Equal count does not mean the sets are necessarily equal.

In equality of ordered pairs, components in the same positions are equal. This rule is very useful in variable-based questions.

((2,2)) is correct because the first (2) is in (A) and the second (2) is in (B). Components may be equal as long as membership is correct.