Cartesian product of sets

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

Explanation: In (B\times A), the first component is (2) from (B) and the second is (1) or (2) from (A). Reversing the order changes the answer.

Correct answer: A. (n(A\times B)=n(B\times A))

Explanation: Both counts are (n(A)n(B)), so the numbers are equal. Remember that counts may be equal even when the sets are different.

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

Explanation: Both elements of (A) come in the first position and (0) from (B) comes in the second position. A negative number is used like any other element.

Correct answer: C. ((6,2))

Explanation: In ((6,2)), (6\in A) and (2\in B), so it is correct. Having both numbers present is not enough; their positions must also be correct.

Correct answer: B. ((1,0))

Explanation: In ((1,0)), both components are from (A), so it belongs to (A\times A). In (A\times A), elements of (A) fill both positions.

Correct answer: A. (A\times C\subset B\times C)

Explanation: If every element of (A) is in (B), then every pair of (A\times C) is also in (B\times C). In subset questions, focus on the first component.

Correct answer: A. (\varnothing)

Explanation: There is no element in (C) for the second component, so no ordered pair is formed. Cartesian product with an empty set is empty.

Correct answer: A. ((a,b))

Explanation: An ordered pair is written as ((a,b)), while ({a,b}) is a set. In exams, notice the difference between parentheses and braces.

Correct answer: A. ((a,b)) and ((b,a)) are generally different

Explanation: The first and second positions in an ordered pair have different meanings. Therefore, do not generally treat ((a,b)) and ((b,a)) as equal.

Correct answer: B. (mn)

Explanation: In Cartesian product, each element of (A) pairs with every element of (B), so the total count is (mn). For counting questions, multiply, do not add.

Correct answer: A. ({(1,3),(1,4),(2,3),(2,4)})

Explanation: In (A\times B), (1,2) from (A) come first and (3,4) from (B) come second. The full list should have (2\times 2=4) pairs.

Correct answer: C. 3

Explanation: The governing concept is the definition of a Cartesian product. For sets A and B, A×B consists of all ordered pairs (a,b) in which the first component belongs to A and the second component belongs to B. Here A={1,2}, so x may be 1 or 2. The second set is the singleton B={3}, so every pair must have y=3. Explicitly, {1,2}×{3}={(1,3),(2,3)}. Since the second coordinate is 3 in both possible pairs, option C is correct. Options A and B are possible values of x, not y. Option D lists the alternatives for the first component and therefore confuses the positions in an ordered pair.

Correct answer: A. (4) or (5)

Explanation: The first component comes from the first set ({4,5}), so (x) can be (4) or (5). Do not mix the first and second components.

Correct answer: C. (3)

Explanation: When the first component (1) is fixed, the second component can be any of the (3) elements of (B). With a fixed first component, the number of pairs is (n(B)).

Correct answer: B. (6)

Explanation: (n(Y)=3) and (n(X)=2), so (n(Y\times X)=3\times 2=6). The same multiplication rule applies even when set names change.

Correct answer: A. ({(a,b_1),(a,b_2)})

Explanation: The element (a) of (A) stays in the first position and both elements of (B) come in the second position. The same rule works for letter elements too.

Correct answer: B. (4)

Explanation: The direct answer is option B, 4. The Cartesian product A × B contains every ordered pair (a,b), so its number of pairs is n(A × B) = n(A) × n(B). Here 8 = n(A) × 2. Dividing both sides by 2 gives n(A) = 8 ÷ 2 = 4. Option A, 2, is not correct because it would give only 2 × 2 = 4 pairs. Option B, 4, is correct because 4 × 2 = 8. Option C, 6, would give 12 pairs, not 8. Option D, 10, would give 20 pairs. Thus the number of elements in A is 4. Remember: for finite sets, multiply the numbers of elements to count Cartesian-product pairs; to find a missing number, divide.

Correct answer: A. ((a,b)\in A\times B\Rightarrow a\in A,\ b\in B)

Explanation: The basic condition of Cartesian product is that the first component is from (A) and the second from (B). Remember this statement like a definition.

Correct answer: C. (3)

Explanation: When the second component (4) is fixed, the first component can be any of the (3) elements of (A). With a fixed second component, the count is (n(A)).

Correct answer: B. (2)

Explanation: The first component (2) is fixed and the second component is chosen from the (2) elements of (B). Therefore, there are (2) such pairs.

Correct answer: B. No, because (3\notin A)

Explanation: The first component of ((3,1)) is (3), but (3\notin A). A correct second component alone does not make the whole pair correct.

Correct answer: A. (A\times B={(0,1)}) and (B\times A={(1,0)})

Explanation: In (A\times B), (0) is first and (1) is second, while in (B\times A) the order is reversed. Order remains important even with one element in each set.

Correct answer: B. (4)

Explanation: For the product of three sets, the count is (n(A)n(B)n(C)), so (2\times 1\times 2=4). This counts ordered triples.

Correct answer: C. (4) options

Explanation: (A) has (2) elements and (B) has (2) elements, so (2\times 2=4) options are formed. Real-life choices can also be counted by Cartesian product.

Correct answer: C. (6) points

Explanation: (n(A\times B)=n(A)n(B)=3\times 2=6), so there are (6) points. Cartesian product can also be understood as coordinate points.