Subset Product/Examples/Subsets of Reals under Multiplication

Example of Subset Product

Let $\struct {\R, \times}$ be the multiplicative group of (non-zero) real numbers.

Let $S = \set {-1, 2}$.

Let $T = \set {1, 2, 3}$.

Then the subset product $S T$ is:

$ST = \set {-1, -2, -3, 2, 4, 6}$

Proof

Taking each ordered pair $\tuple {s, t}$ from $S \times T$:

\(\ds -1 \times 1\)

\(=\)

\(\ds -1\)

\(\ds -1 \times 2\)

\(=\)

\(\ds -2\)

\(\ds -1 \times 3\)

\(=\)

\(\ds -3\)

\(\ds 2 \times 1\)

\(=\)

\(\ds 2\)

\(\ds 2 \times 2\)

\(=\)

\(\ds 4\)

\(\ds 2 \times 3\)

\(=\)

\(\ds 6\)

$\blacksquare$

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Subgroups