Basics of Cartesian Products and Ordered Pairs

The Cartesian product A × B is defined as the set of ordered pairs whose first component comes from A and whose second component comes from B. To see why the other options fail, note that {x | x ∊ A and x ∊ B} describes the intersection A ∩ B (elements that are simultaneously in both sets), not ordered pairs; {a ∪ b | a∊A,b∊B} would produce unions of elements which generally are not ordered pairs; and {a ∩ b | a∊A,b∊B} produces intersections of elements. The ordered-pair description explicitly pairs each a ∈ A with each b ∈ B, which is exactly the product.

By the definition of A × B, take each element of A as the first coordinate and pair it with each element of B as the second coordinate. A has two elements 1 and 2; B has the single element x. The pairs are (1,x) and (2,x). Option A lists reversed pairs (elements of B first), option B lists an unrelated pair and a pointless (x,x), and option D is not a set of ordered pairs.

The Cartesian product A × B contains ordered pairs (a,b) with a ∈ A and b ∈ B. If A is empty there are no possible first coordinates a, so no ordered pairs can be formed. Therefore A × B is empty. It is not equal to B (which has elements), not equal to A as a set identity (though A is ∅, saying A × B = A could misleadingly match ∅ but the reason is the product is empty because A has no elements), and {∅} would be a one-element set containing the empty set, which is different from the empty set itself.

Cardinality of a Cartesian product of finite sets multiplies: |A × B| = |A| · |B| because for each of the 3 choices of a ∈ A there are 4 choices of b ∈ B, giving 3·4 = 12 ordered pairs. Options 7, 1, 0 are arithmetic errors or misunderstandings.

The set {1,2} × {a,b} consists of ordered pairs with first coordinate from {1,2} and second from {a,b}. Thus (1,a) is in the product. (a,1) would be an element of {a,b} × {1,2}, not the given product. {1,a} is a two-element set, not an ordered pair. (1,2) has second coordinate 2, which is not in {a,b}.

Each element of A must pair with each element of B. Both sets contain the single element 0, so the only ordered pair is (0,0). The empty set is wrong because pairs can be formed; {0} is the one-element set containing number 0, not the ordered pair; {(0)} would be a set containing the one-element tuple (0) which is not the ordered pair notation.

The sets A × B and B × A are equal if and only if either A = B or at least one of A or B is empty. When A = B, both products contain exactly the same ordered pairs. However, when both sets are non-empty and unequal, the products differ. For individual elements, (a,b) ≠ (b,a) when a ≠ b, which is why we say Cartesian products are not commutative in general. When A ≠ B, the products are different sets: if a ∈ A\B and b ∈ B, then (a,b) ∈ A × B but (a,b) ∉ B × A.

Distribute A over the union in the second coordinate: an element (a,x) ∈ A × (B ∪ C) iff a ∈ A and x ∈ (B ∪ C), meaning x ∈ B or x ∈ C. So (a,x) is in either A × B or A × C, hence the union. Intersection would correspond to product with intersection of the second factors, (A ∪ B) × C swaps roles incorrectly, and A × B × C denotes triples, not pairs.

An ordered pair (x,c) is in (A ∩ B) × C if and only if x ∈ A and x ∈ B and c ∈ C. That means (x,c) is in both A × C and B × C, so it lies in their intersection. The union would allow x in A or B, not necessarily both. A × (B ∩ C) would require c in B ∩ C, which is a different condition. (A ∩ C) × (B ∩ C) intermixes coordinates incorrectly.

There can be no ordered pair (a,b) with b ∈ ∅ because ∅ has no elements. Therefore the product is empty. It is not equal to A (which has elements), nor {A} (the singleton containing A), nor {∅} (a singleton with empty set).

(A × B) is the set of ordered pairs (1,a),(1,b),(2,a),(2,b). Now take the product of that set with C = {x}; each element of (A × B) becomes the first coordinate and x becomes the second, so the elements are pairs of the form ((a,b),x). Option B lists triples (1,a,x) which is informal shorthand but not the formal nested ordered-pair structure; option C mixes nesting; option D is incomplete.

The elements of (A × B) × C are of the form ((a,b),c) while elements of A × (B × C) are of the form (a,(b,c)). These are not the same ordered-pair objects, so as sets they are not literally equal. However there is a natural bijection φ((a,b),c) = (a,(b,c)) between them, so they are structurally the same in the sense of cardinality and correspondence. Finiteness is irrelevant to the equality issue.

For each choice of a ∈ A there are |B| choices of b and |C| choices of c, so by the rule of product there are |A|·|B|·|C| distinct ordered triples (or, formally, nested ordered pairs) in the product. Summing or taking max or mixing plus is incorrect for counting the product set.

Equality of ordered pairs is defined componentwise: (a,b) equals (c,d) exactly when the first coordinates are equal and the second coordinates are equal. The other options misstate or weaken the condition.

ℤ × ℤ is the set of ordered pairs of integers. (1,2) is such a pair. {1,2} and {1} are sets, not ordered pairs; 3 is an integer, not a pair.

B × A contains ordered pairs with first coordinate from B and second from A. (1,a) has first coordinate 1 which is in A, not B, so it is in A × B but not in B × A. It would be in B × A only if the first component belonged to B.