Writing Proofs, Part III - quantifiers

A quantifier is a phrase that puts a variable into context. The most common ones are "There is a ..." and " For all ..."

In these examples, the variable will be x, which stands for a real number. In general, the "variable" could really be a function, a matrix, or whatever.

Example 1: For all x, (x + 1) + 1 = x + 2.
Example 2: There is an x such that x + 1 =5.
Example 3: If the quantifier is missing, the sentence is (slightly) wrong, but probably "for all" was intended. On the previous page I wrote "If x>2 then x>3" but really meant "For all x, if x>2 then x>3". In practice, the "for all" phrase is pretty similar to "if-then". For example, you could say "For all x>2, x>3".

Let's start with some True-False practice involving quantifiers. As before, think about why you are answering True or False, and how you'd explain your answer to a friend.

 

True False. For all x, if x >0, then 3x > 0.

True False. For all x, x + 12 = 5.

True False. There exists an x, such that x + 12 = 5.

True False. For all x, 2(x+1) = 2x+2.

True False. There is an x, 2(x+1) = 2x+2.

True False. There is an x, such that x > 5 and 3x + 1 < 10.

(try again, until you get all 6)

Comments: Ex1 is true, and similar to an example on the previous page. Ex 2 is false, as you can see from a single counterexample, such as x = 0. Ex 3 is true when x = -7. Ex 4 is true. The explanation should be based on general principles (such as the distributive property). Ex 5 is true, and a simple example like x = 0 is a proof (or you could use Ex 4). Ex 6 is false. You can use algebra to show the two parts contradict each other.

Summary of these proof strategies:

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

Suppose you want to prove that "For all x, if x >0, then 3x > 0." is true. Then you'd (choose one):

Assume x >0, then prove that 3x > 0 using general principles.

Find an example of an x, with x>0 and 3x>0.

You want to prove that " For all x, x + 12 = 5." is false. You'd

Find an example of an x, such that x + 12 = 5 is true.

Find an example of an x, such that x + 12 = 5 is false.

You want to prove that " There exists an x, such that x + 12 = 5 " is true. You'd

Find an example of an x, such that x + 12 = 5 is true.

Find an example of an x, such that x + 12 = 5 is false.

You want to prove that "For all x, 2(x+1) = 2x+2. " is true. You'd

Give a general argument, without choosing x.

Point out that pigs cannot fly.

To prove that "There is an x, 2(x+1) = 2x+2. " is true, you'd

Find an x so that 2(x+1) = 2x+2.

Assume that 2(x+1) and prove that 2x+2.

To prove that "There is an x, such that x > 5 and 3x + 1 < 10 " is false, you'd

Assume there is such an x, and show that leads to a contradiction.

Find an example of an x so that x>5 is false and 3x+1 < 10 is false.

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 IV but it's not written yet, and probably won't be, unless you ask for it. I'd like to know whether this kind of drill really helps you.

Written by S.Hudson, 5/1/02

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