Choose
arbitrary sets [math]A
[/math],
[math]B
[/math], and
[math]C
[/math]. We
want to show to show that
every [math]x \href{/cs2800/wiki/index.php/%E2%88%88}{∈} A \href{/cs2800/wiki/index.php/%E2%88%A9}{∩} (B \href{/cs2800/wiki/index.php/%E2%88%AA}{∪} C)
[/math] is also
in [math](A \href{/cs2800/wiki/index.php/%E2%88%A9}{∩} B) \href{/cs2800/wiki/index.php/%E2%88%AA}{∪} (A \href{/cs2800/wiki/index.php/%E2%88%A9}{∩} C)
[/math]. Choose an
arbitrary [math]x
[/math] in the
left hand side. Then
[math]x ∈ A
[/math] and [math]x ∈ B \href{/cs2800/wiki/index.php/%E2%88%AA}{∪} C
[/math]. This means either (1)
[math]x ∈ B
[/math] or
(2)
[math]x ∈ C
[/math]. In
case (1), we have
[math]x \href{/cs2800/wiki/index.php/%E2%88%88}{∈} A
[/math] and
[math]x \href{/cs2800/wiki/index.php/%E2%88%88}{∈} B
[/math], so
[math]x \href{/cs2800/wiki/index.php/%E2%88%88}{∈} A \href{/cs2800/wiki/index.php/%E2%88%A9}{∩} B
[/math], and thus
[math]x \href{/cs2800/wiki/index.php/%E2%88%88}{∈} \href{/cs2800/wiki/index.php/RHS}{RHS}
[/math]. Case (2) is similar. In either case,
[math]x \href{/cs2800/wiki/index.php/%E2%88%88}{∈} RHS
[/math], as required.
