Last modified on 16 June 2009, at 11:53

Mathematical Proof/Appendix/Answer Key/Mathematical Proof/Methods of Proof/Constructive Proof

Problem 1.1) First, we wish to show that A\cup(B\cap C) \subset (A\cup B)\cap(A\cup C). Let x \in A\cup(B\cap C). Then x \in A or x \in B\cap C.

case 1: x \in A
x \in A \subset A\cup B so that x \in A\cup B
x \in A \subset A\cup C so that x \in A\cup C
x \in A\cup B and x \in A\cup C so that x \in (A\cup B)\cap (A\cup C)

case 2: x \in B\cap C
x \in B and x \in C
x \in B \subset A\cup Bso that x \in A\cup B
x \in C \subset A\cup C so that x \in A\cup C
x \in A\cup B and x \in A\cup C so that x \in (A\cup B)\cap (A\cup C)

Since in both cases, x \in (A\cup B)\cap (A\cup C), we know that A\cup(B\cap C) \subset (A\cup B)\cap(A\cup C)

Now we wish to show that (A\cup B)\cap(A\cup C) \subset A\cup(B\cap C). Let x \in (A\cup B)\cap (A\cup C). Then x \in A\cup B and x \in A\cup C.

case 1a: x \in A
x \in A \subset A\cup(B\cap C) so that x \in A\cup(B\cap C)

case 1b: x \in B
We can't actually say anything we want to with just this, so we have to also add in x \in A\cup C
case 2a: x \in A : SEE CASE 1a
case 2b: x \in C
We now have x \in B and x \in C so that x \in B\cap C
Of course, since x \in B\cap C \subset A\cup(B\cap C), it follows that x \in A\cup(B\cap C).
Since both cases 2a and 2b yield x \in A\cup(B\cap C), we know that it follows from 1b.

Since in both cases 1a and 1b, x \in A\cup(B\cap C), we know that (A\cup B)\cap(A\cup C) \subset A\cup(B\cap C).

Since both A\cup(B\cap C) \subset (A\cup B)\cap(A\cup C) and (A\cup B)\cap(A\cup C) \subset A\cup(B\cap C), it follows (finally) that A\cup(B\cap C) = (A\cup B)\cap(A\cup C).


--will continue later, feel free to refine it if you feel it can be--