Cartesian product of sets

Correct answer: C. (4)

Explanation: The condition \\(|a-b|=3\\) means that the two numbers differ by exactly three, without regard to which one is larger. Since both A and B contain the numbers 1 through 5, we can list the pairs directly. With a smaller first value, the possibilities are (1,4) and (2,5). Reversing each pair gives (4,1) and (5,2).

These four ordered pairs all belong to A × B and satisfy the absolute-difference condition. No other pair works: differences involving adjacent or closer numbers are smaller than 3, and larger gaps are not available within 1 to 5. Because ordered pairs are different when their order is reversed, all four must be counted. Thus option C is correct.

Correct answer: B. 10

Explanation: The sum \(a+b\) is even exactly when both \(a\) and \(b\) are even or both are odd. Set \(A\) has 3 even and 2 odd elements, while set \(B\) has 2 even and 2 odd elements. Thus, the number of even-even pairs is \(3\times2=6\), and the number of odd-odd pairs is \(2\times2=4\). Therefore, the total is \(6+4=10\), so option B is correct. The distractor 12 usually results from miscounting one parity group. Exam tip: for an even sum, count the even-even and odd-odd cases separately.

Correct answer: A. (18)

Explanation: For (a=1,2,3,4,5,6), the counts of (b) are (4,4,4,3,2,1), totaling (18). Include equality in (\le).

Correct answer: C. (6)

Explanation: The product is odd only when both components are odd. (A) has (3) odd elements and (B) has (2), so (3\cdot2=6).

Correct answer: C. 4

Explanation: For each a ∈ A, evaluate b = 3a. For a = 1, 2, 3, and 4, the corresponding values of b are 3, 6, 9, and 12, all of which belong to B. For a = 5, b = 15, which is not in B. Thus the valid ordered pairs are (1, 3), (2, 6), (3, 9), and (4, 12), giving a total of 4 pairs. Exam tip: Count only those values of a for which 3a is an element of B.

Correct answer: C. 4

Explanation: From \(a-2b=1\), we get \(a=2b+1\). Substituting \(b=0,1,2,3\) gives \(a=1,3,5,7\), respectively, and all these values belong to \(A\). Thus the valid ordered pairs are \((1,0),(3,1),(5,2),(7,3)\), giving a total of 4 pairs. Exam tip: test every value of one set in the equation and retain only those for which the resulting value belongs to the other set.

Correct answer: C. 10

Explanation: The difference of two integers is even exactly when both integers have the same parity—both even or both odd. Set \(A\) has 2 even and 2 odd numbers, while set \(B\) has 2 even and 3 odd numbers. Therefore, the number of valid ordered pairs is \(2\times2+2\times3=4+6=10\). The pairs are ordered, so the first element is selected from \(A\) and the second from \(B\). Exam tip: for an even difference, count pairs with matching parity.

Correct answer: C. 30

Explanation: The governing concept is complement counting in a Cartesian product. Since A and B each contain six elements, the total number of ordered pairs in A × B is 6 × 6 = 36. It is easier to count the complement of the required condition. The pairs that do not satisfy a ≠ b are exactly those with a = b: (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6). There are six such diagonal pairs. Therefore the number satisfying a ≠ b is 36 − 6 = 30, so option C is correct. Option D counts every pair and ignores the restriction, whereas options A and B result from subtracting an incorrect number of equal pairs. Ordered-pair order is preserved throughout.

Correct answer: D. (7)

Explanation: For every (a\in A), (a^2) is in (B), so (7) pairs are formed. Watch squares of negative numbers carefully.

Correct answer: B. (10)

Explanation: Checking all sums gives (10) pairs with prime sum. In such questions, test possible sums systematically.

Correct answer: C. (3)

Explanation: The condition b divided by a equals 4 means that b must be four times a. A Cartesian product contains ordered pairs whose first member comes from A and whose second member comes from B. Thus, we must test every value of a in A and calculate its required value of b. This makes the condition easy to check without listing all pairs of the product.

For a=1, b=4, and 4 belongs to B, giving (1,4). For a=2, b=8, giving (2,8). For a=4, b=16, giving (4,16). All three required second components are in B, so there are exactly 3 suitable ordered pairs. Therefore option C is correct. A count of 1 or 2 misses valid pairs, while 4 is impossible because A has only three possible first components.

Correct answer: C. (5)

Explanation: The satisfying pairs are ((4,4),(4,5),(5,4),(5,5)), so the total is (4). Count each ordered pair once.

Correct answer: B. (4)

Explanation: The pairs with sum divisible by (5) are ((1,4),(2,3),(3,2),(4,1)). Counting by remainders is easier.

Correct answer: B. 13

Explanation: The governing concept is systematic counting of ordered pairs subject to divisibility. Fix each a in A and list its positive multiples that also belong to B. For a = 1, all six members of B qualify: 1, 2, 3, 4, 5, 6, giving six pairs. For a = 2, the qualifying multiples are 2, 4, 6, giving three. For a = 3, they are 3 and 6, giving two. For a = 4, only 4 belongs to B, giving one, and for a = 5, only 5 qualifies, giving one. The total is 6 + 3 + 2 + 1 + 1 = 13. Hence option B is correct. A number such as 8 cannot be used because it is outside B.

Correct answer: B. {1,3,5}

Explanation: In a Cartesian product A × C, every ordered pair has its first component from A and its second component from C. The first components shown are 2 and 4, agreeing with A. The second components appearing in the pairs are 1, 3 and 5, so C = {1,3,5}. This also produces all combinations: each of the two elements of A is paired with each of the three elements of C, giving 2 × 3 = 6 pairs, exactly as listed. Therefore option B is correct. Option A confuses the first component with the second, option C includes elements that never occur as second components, and option D is a set of pairs rather than the required set C.

Correct answer: A. (B=\varnothing)

Explanation: Since (A) is non-empty and (A\times B=\varnothing), we must have (B=\varnothing). An empty product means at least one set is empty.

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

Explanation: Order inside a set does not matter, so (A=B), and both products are equal. First check whether the two sets are actually the same.

Correct answer: B. 7

Explanation: Here, \(|A|=4\). For the Cartesian product, \(|A\times B|=|A|\cdot|B|\). Thus, \(28=4\cdot|B|\), giving \(|B|=28/4=7\). Therefore, the correct answer is 7. Exam tip: the number of ordered pairs in a Cartesian product equals the product of the cardinalities of the two sets.

Correct answer: C. (4095)

Explanation: The direct answer is option C, 4095. The product A × B has n(A × B) = 3 × 4 = 12 elements. For any set with n elements, each element has two choices in a subset—present or absent—so the total number of subsets is 2^n. Here that is 2^12 = 4096. Exactly one of these is the empty subset, so the number of non-empty subsets is 4096 − 1 = 4095. Option A, 255, equals 2^8 − 1 and uses the wrong product size. Option B, 1023, equals 2^10 − 1 and is also based on an incorrect exponent. Option C, 4095, uses 12 correctly and removes the empty subset. Option D, 8191, equals 2^13 − 1, so it is one exponent too large. The key sequence is multiply first, use 2^n, then subtract 1 for “non-empty.”

Correct answer: C. (4)

Explanation: The condition gives ((1,4),(2,3),(3,2),(4,1)), so there are (4) elements. A relation is a subset of (A\times B).

Correct answer: B. Always odd

Explanation: The sum of an even number and an odd number is always odd. In parity questions, listing all pairs is not necessary.

Correct answer: A. (14)

Explanation: For (a=0,1,2,3), the counts of (b) are (5,4,3,2), totaling (14). In a strict inequality, the boundary is not included.

Correct answer: B. 3

Explanation: From \(a+b=11\), we get \(b=11-a\). Checking the elements of \(A\), the valid ordered pairs are \((1,10),(3,8),(5,6)\), and each second component belongs to \(B\). Thus, there are 3 pairs. Exam tip: Since 11 is odd and every element of \(B\) is even, \(a\) must be odd; therefore, only \(1,3,5\) need to be checked.

Correct answer: C. (10)

Explanation: Direct answer: Option C, 10. A relation here contains ordered pairs satisfying \\(a\\le b\\), where both numbers are from 1, 2, 3, 4. For a=1, all four b-values work: 4 pairs. For a=2, b=2,3,4 work: 3 pairs. For a=3, b=3,4 work: 2 pairs. For a=4, only b=4 works: 1 pair. The total is \\(4+3+2+1=10\\). Option A, 6, misses several pairs; B, 8, also undercounts; C is correct. D, 12, overcounts pairs that do not satisfy the condition. The equality sign is important: \\(a=b\\) is allowed because the symbol is \\(\\le\\), not just <. Memory cue: list each row from the diagonal to the right.

Correct answer: C. (9)

Explanation: Independent calculation shows a problem: the supplied answer C=9 is not correct. Count multiples in B for each a. For a=2, the values 4, 6, 10, and 12 are divisible by 2, giving 4 pairs. For a=3, 6, 12, and 15 work, giving 3 pairs. For a=5, 10 and 15 work, giving 2 pairs. Therefore the total is \\(4+3+2=9\\), actually matching Option C. Option A says 7 and undercounts; B says 8 and also undercounts; C is correct because all valid pairs are included; D says 10 and overcounts. The supplied short explanation is accurate: count multiples separately for each a. The wording is clear, so no correction is needed.