Subset Product/Examples/Subset Product with Empty Set

Example of Subset Product

Let $\struct {S, \circ}$ be an algebraic structure.

Let $A, B \in \powerset S$.

If $A = \O$ or $B = \O$, then $A \circ B = \O$.

Proof

We show the contrapositive:

Suppose that $A \circ B \ne \O$.

Then:

$\exists x: x \in A \circ B$

From definition of subset product:

$A \circ B = \set {a \circ b: a \in A, b \in B}$

Therefore:

$\exists a \in A, b \in B: x = a \circ b$

In particular, both $A$ and $B$ are non-empty.

By De Morgan's Laws (Logic)/Conjunction of Negations, it is not the case that $A = \O$ or $B = \O$.

Hence the result by Proof by Contraposition.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 41$: Multiplication of subsets of a group