Cartesian Product of Sets
Question 1:
Find $A \times B$, $A \times A$ and $B \times A$ given that:
(i) $A=\{2,-2,3\}$ and $B=\{1,-4\}$
Solution:
The Cartesian product of two sets $X$ and $Y$, denoted as $X \times Y$, is the set of all ordered pairs $(x,y)$ such that $x \in X$ and $y \in Y$.
Step 1: Find $A \times B$
To find $A \times B$, we pair every element in set $A$ with every element in set $B$. The first element of each pair must come from $A$, and the second from $B$.
$A \times B = \{(2,1), (2,-4), (-2,1), (-2,-4), (3,1), (3,-4)\}$
Step 2: Find $A \times A$
To find $A \times A$, we pair every element in set $A$ with every other element in set $A$ (including itself).
$A \times A = \{(2,2), (2,-2), (2,3), (-2,2), (-2,-2), (-2,3), (3,2), (3,-2), (3,3)\}$
Step 3: Find $B \times A$
To find $B \times A$, we pair every element in set $B$ with every element in set $A$. Notice that the order is reversed compared to step 1; the first element must now come from $B$.
$B \times A = \{(1,2), (1,-2), (1,3), (-4,2), (-4,-2), (-4,3)\}$