Math 2513
Example 10, page 89
In this course students are strongly encouraged to learn to write mathematical proofs using
everyday, common-sense language, and not relying on the use of arcane logical symbols. Here
is a proof of the second of DeMorgan’s laws which avoids using logical symbols. Compare
this with the proof of Example 10 on page 89 of Rosen’s book.
Example 10.Prove that A∩B=A∪B.
proof: We will prove that these two sets are equal by showing that each is a subset of the
other.
Suppose that xis an element of A∩B. By the definition of complementation this means
that x /∈A∩B. For xto be an element of A∩Bwe must have x∈Aand x∈B. So,
since xis not an element of A∩Bthen either x /∈Aor x /∈B. Using the definition of
complementation, this means that x∈Aor x∈B. Therefore x∈A∪Bby the definition of
union. This shows that A∩B⊆A∪B.
Now suppose that x∈A∪B. From the definition of union it follows that x∈Aor x∈B.
Thus x /∈Aor x /∈Bby the definition of complementation. If xwere an element of A∩B
then xwould be an element of both Aand Bwhich is impossible. So we conclude that
x /∈A∩B. By the definition of complementation this means that x∈A∩B. This shows
that A∪B⊆A∩B. Since we have shown that each set is a subset of the other, the two sets
are equal, and DeMorgan’s second identity is proved.