Writing Proofs, Part I - "and" and "or"

Let's start with some True-False practice involving easy compound sentences. They all contain the word "and" or the word "or". As you go, think about why you are answering True or False, and how you'd explain your answer to a friend.

Comments: In the "and" examples, every part must be true (as in ex 3) and you'd have to explain the parts to your friend one at a time. If even one part is false, you answer false. You'd explain to your friend why that one part is false (ex 2 is false, because 125 is not even).

In the "or" examples, at least one part should be true. In ex 1, you'd explain that 125 is odd, so the answer is True. To answer False, all parts must be false. Ex 4 is false because 125 is not negative and not prime.

Summary of these proof strategies:

• to prove "p and q" is true, you prove them one at a time. It's like two separate proofs.
• to prove "p and q" is false, you pick one (wisely) and prove it is false.
• to prove "p or q" is true, you pick one (wisely) and prove it is true.
• to prove "p or q" is false, you prove both are false, one at a time.

Saying that "p or q" is false is the same as saying that "p is false" and "q is false". Knowing which statements are equivalent can simplify your life. OK, let's practice with these ideas a bit.

Suppose you want to prove that "125 is either an even number or an odd number" is true. Then you'd (choose one):

If you feel lost and want to do some background reading, contact me. Or if you just want to tell me what you think of this page. If you are ready to go on, you can try Writing Proofs, Part II ("if - then").

Written by S.Hudson, 5/1/02

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