Cartesian Product of Sets
Q: Find $A \times B$, $A \times A$ and $B \times A$ given that $A = B = \{p, q\}$.
Solution:
First, let's write down the given sets:
$A = \{p, q\}$
$B = \{p, q\}$Step 1: Finding $A \times B$
To find the Cartesian product, we pair each element of set $A$ with each element of set $B$.
$$A \times B = \{(p, p), (p, q), (q, p), (q, q)\}$$
Step 2: Finding $A \times A$
We do the same thing, but pairing set $A$ with itself. Since $A$ and $B$ are identical sets, the result will be exactly the same.
$$A \times A = \{(p, p), (p, q), (q, p), (q, q)\}$$
Step 3: Finding $B \times A$
Here, we take the first element from $B$ and the second from $A$. Because $A = B$, swapping the order changes nothing.
$$B \times A = \{(p, p), (p, q), (q, p), (q, q)\}$$
(Ans.)