I'm currently doing work on discrete mathematics in my free time and am having some difficulties with understanding some questions pertaining to Relations and Functions. To be specific, I'm stuck on the following practice question:
Let A = {a} and B = {1, 2}. Find each of the following:
a. A × B
b. P(A × B)
c. P(A) and P(B)
d. P(A) × P(B)
Where P() is defined as a powerset.
How would I go about finding the results of A and B to satisfy these relations? Any help is greatly appreciated! Thanks.
Hint
$A \times B = \{ (a,b) : a \in A \ \text {and} \ b \in B \}$.
Thus, $A \times B = \{ (a,1), (a,2) \}$.
The set $\mathcal P (A \times B)$ is the set of all subsets of $A \times B$. In general, if a set $M$ has $n$ elements, its power-set $\mathcal P(M)$ has $2^n$ elements.
Thus, due to the fact that $A \times B$ has only $2$ elements, we have that $\mathcal P (A \times B)$ has $2^2=4$ elements; they are the four subsets of $A × B$, i.e. $∅,\{ (a,1) \}, \{ (a,2) \}$ and $A×B$ itself.