1. **Problem:** Find the Cartesian product $A \times B$ where $A = \{1,2,3\}$ and $B = \{a,b,c\}$, defined as $A \times B = \{(a,b) : a \in A, b \in B\}$.\n\n2. **Formula and Explanation:** The Cartesian product $A \times B$ consists of all ordered pairs where the first element is from set $A$ and the second element is from set $B$.\n\n3. **Step-by-step Solution:**\n- Take each element from $A$: 1, 2, 3.\n- Pair it with each element from $B$: a, b, c.\n- This gives pairs: $(1,a), (1,b), (1,c), (2,a), (2,b), (2,c), (3,a), (3,b), (3,c)$.\n\n4. **Final answer:**\n$$A \times B = \{(1,a), (1,b), (1,c), (2,a), (2,b), (2,c), (3,a), (3,b), (3,c)\}$$\n\n---\n\n1. **Problem:** Find the Cartesian product $A \times B$ where $A = \{1,2,3\}$ and $B = \{a,b,c\}$, but defined as $A \times B = \{(a,b) : a \in B, b \in A\}$.\n\n2. **Formula and Explanation:** Here, the roles of $A$ and $B$ are swapped in the ordered pairs. The first element comes from $B$ and the second from $A$.\n\n3. **Step-by-step Solution:**\n- Take each element from $B$: a, b, c.\n- Pair it with each element from $A$: 1, 2, 3.\n- This gives pairs: $(a,1), (a,2), (a,3), (b,1), (b,2), (b,3), (c,1), (c,2), (c,3)$.\n\n4. **Final answer:**\n$$A \times B = \{(a,1), (a,2), (a,3), (b,1), (b,2), (b,3), (c,1), (c,2), (c,3)\}$$