Writing Proofs, II

Writing Proofs, Part II - "if-then"

This is a little trickier than the previous page. Let's start with some True-False practice involving easy "if-then" sentences. As before, think about why you are answering True or False, and how you'd explain your answer to a friend.

HINT 1: "if p then q" is equivalent to "p is false or q is true".

HINT 2: The last two are supposed to be true for all possible numbers x. If you can find an x that makes "p" true and "q" false, you should answer False.

Comments: In the first four exs, the only combination that you should call False is the "If True, then False" one (ex 4). Ex 1 and ex 2 are true because "pigs can fly" is false. Ex 3 is true because "cats cannot fly" is true.

Ex 5 is false because x might be 2.5, for example, which produces the If True then False combo.

Ex 6 is true. You can assume x > 3 in your reasoning. Use the transitive property and the fact that 3 > 2 to deduce that x > 2.

Summary of these proof strategies:

to prove "if p then q" is true, you can prove p is false or that q is true.
to prove "if p then q" is false, you prove that p is true and then that q is false.
if there are variables like "x", you probably will have to change strategies. To prove it false, you find a counterexample (like x = 2.5 above). To prove it true, you assume p and then work on proving q.

OK, let's practice with these ideas a bit.

Suppose you want to prove that "If pigs can fly then cats can fly" is true. Then you'd (choose one):

point out that pigs cannot fly.

point out that cats cannot fly.

You want to prove that "If pigs can fly then cats cannot fly" is true. You'd (choose one)

point out that that pigs cannot fly [or point out that cats cannot fly].

point out that cats are lighter than pigs.

You want to prove that "If pigs cannot fly then cats cannot fly" is true. You'd

point out that pigs cannot fly

point out that cats cannot fly

You want to prove that " If pigs cannot fly then cats can fly " is false. You'd

point out that pigs cannot fly and then that cats cannot fly.

just point out that pigs cannot fly.

To prove that "If x > 2 then x > 3 " is false, you'd

Assume x>2 and prove that "x>3" is false.

Find a number x > 2, that makes "x>3" false.

To prove that "If x > 3 then x > 2 " is true, you'd

Assume x>3 and prove that x>2.

Find an example of a number bigger than 2 and bigger than 3.

Repeat until you are getting these right.

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 III (quantifiers).

Written by S.Hudson, 5/1/02

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