Thread: Cartesian product

Prove: If $\displaystyle A \subset C $ and $\displaystyle B \subset D$, then $\displaystyle A \times B \subset C \times D$.

Originally Posted by onemore

Prove: If $\displaystyle A \subset C $ and $\displaystyle B \subset D$, then $\displaystyle A \times B \subset C \times D$.

Note that $\displaystyle A \times B = \{ (a,b) : a \in A, b \in B \}$ and $\displaystyle C \times D = \{ (c,d) : c \in C, d \in D \}$

So, pick an element in $\displaystyle A \times B $, say (a,b). You want to show that it is in $\displaystyle C \times D $.

Now, as $\displaystyle a \in A \subset C $, we have that $\displaystyle a \in C $

Can you take it from here?