Worksheets — Ordered Pairs & Cartesian Products

Find $x$ and $y$ for the following equal ordered pairs:
(a) $(x + 2, y - 5) = (6, 2)$     (b) $(3x, 2y + 1) = (9, 7)$

Let $A = \{a, b\}$ and $B = \{1, 2, 3\}$. List all elements of $A \times B$ and $B \times A$.

If $n(A) = 5$ and $n(B) = 6$, find $n(A \times B)$ and $n(B \times A)$. Are they equal?

If $n(A) = 3$ and $n(B) = 4$, how many subsets does the Cartesian product $A \times B$ have?

Let $A = \{1, 2\}$, $B = \{2, 3\}$, and $C = \{3, 4\}$. Verify that:
$A \times (B \cup C) = (A \times B) \cup (A \times C)$

Let $A = \{1, 2, 3\}$ and $B = \{3, 4, 5\}$. Find the intersection set $(A \times B) \cap (B \times A)$.

Prove algebraically that for any sets $A, B, C$, and $D$:
$(A \times B) \cap (C \times D) = (A \cap C) \times (B \cap D)$

If $(x^2 - 1, y^3 + 2) = (8, 10)$, find all possible real values of $x$ and $y$.

Let $A \times B$ have 6 elements. If three elements of $A \times B$ are $(1, x)$, $(2, y)$, and $(3, x)$, find the complete sets $A$ and $B$, and list the remaining elements of $A \times B$.