Question

If $P=\{a,b,c\}$ and $Q=\left\{r\right\}$, form the sets $P\times Q$ and $Q\times P$. Are these two products equal.

Answer 1

Given: $P=\{a,b,c\}$ and $Q=\left\{r\right\}$
The cartesian product of two non-empty sets $A$ and $B$ is given as
$A\times B=\left\{\right(a,b):a\in A,b\in B\}$
So,
$P\times Q=\left\{\right(a,r),(b,r),(c,r\left)\right\}$
$Q\times P=\left\{\right(r,a),(r,b),(r,c\left)\right\}$
Since, by the definition of equality of ordered pairs, the pair $(a,r)$ is not equal to the pair $(r,a)$, we conclude that $P\times Q\ne Q\times P$