Math Help -> Logic and Proof -> Truth Value

Truth Value

Peter writes,

Dear Graeme,

I notice from your web site that you welcome requests for assistance. I hope that I am not being too bold in giving you this long request, here goes:

I would like to find out about the symbols ==> and <==> as used in say a delta epsilon proof; Take for example:

|x^2-4| < epsilon
<==> |(x+2)^2 - 4*(x+2)| < epsilon
<== |x+2|^2 + 4*|x+2| < epsilon (by the triangle inequality)
<==> |x+2|^2 + 4*|x+2| - epsilon < 0

does this read from bottom to top in everyday english as:

IF (|x+2|^2 + 4*|x+2| - epsilon) when evaluated equals a negative quantity

THEN |x^2-4|when evaluated would be less than epsilon?

And does it mean that read from top to bottom in everyday english as:

IF |x^2-4| when evaluated is less than epsilon?
THEN
(|x+2|^2 + 4*|x+2| - epsilon) when evaluated does not have to equal a negative quantity?

Regarding the three lines:

(1)    <==> |(x+2)^2 - 4*(x+2)| < epsilon
(2)    <== |x+2|^2 + 4*|x+2| < epsilon (by the triangle inequality)
(3)    <==> |x+2|^2 + 4*|x+2| - epsilon < 0

Are these the correct interpretations?

If line 1 is true line 2 is not necessarly true
If line 2 is true line 1 is true
If line 2 is true line 3 is true
If line 3 is true line 2 is true

I have found these explanations on the internet but I am having difficulty understanding and applying them to the above case

Conditional
If ....then p ==> q (if p, then q)

p q p ==> q

T T T

T F F

F T T

F F T

p ==> q is only false if p is true and q is false

Biconditional
If and only if p <==> q ( p if and only if q)

p q p <==> q

T T T

T F F

F T F

F F T

p <==> q is true only when both p and q have the same truth value. That means when both are true or both are false.

I had assumed that p was the statement that was to be tested and q is the result depending if p is true or false. From looking at the above truth tables I realize that my assumption is incorrect but how do I know what to call what? In fact I am so mixed up now that I would find it difficult to even guess what is what.

Could you please show me how to apply the above truth tables to the Regarding the three lines:
(1)    <==> |(x+2)^2 - 4*(x+2)| < epsilon
(2)    <== |x+2|^2 + 4*|x+2| < epsilon (by the triangle inequality)
(3)    <==> |x+2|^2 + 4*|x+2| - epsilon < 0

For instance what are the p and q terms?

Best regards and thanks in advance,
Peter

First, you are not being too bold. I welcome requests such as this.

Second, the symbol ==> means "implies". For example, if p is "Joe is happy" and q is "Joe whistles", then p==>q means p implies q, or "Joe is happy" implies "Joe whistles". Phrased as an if-then statement, p==>q becomes: If Joe is happy then Joe whistles.

The symbol <==> means "implies and is implied by". Phrased as an if-then statement, it becomes p if and only if q, which is sometimes stated as p iff q. Joe is happy if and only if Joe whistles.

I rarely see the symbol <==, but it would be totally consistent to define it to mean "is implied by". Phrased as an if-then statement, p<==q would mean p if q. (I suspect this symbol is rarely used, because it doesn't help form proofs, which rely on a sequence of statements, such that each one is true if the one before it is true.)

In proofs, the symbols ==> or <==> are helpful because you can start with a true statement, p1, and derive a sequence of true statements p2, p3, etc. to arrive at a conclusion, which must also be true, as in

p1 ==> p2 <==> p3 ==> q

This is a proof of the statement "if p1 then q". In proofs of this kind we only consider the "True" truth value of the initial statement, p1, and we assume the truth value of each statement is also "True".

Even in a proof of the statement p1 iff q, the two legs of the proof are typically treated separately. That is, a typical proof of p1 iff q would be divided into if q then p1 (the "if" part) and if p1 then q (the so-called "only if" part). That's because it is rare to find a proof in which every step is a two-way implication.

In algebra, too, we typically assume that the statements we are given are true. For example, we are told that

x^2-x-2=0, and asked to find x. So we write this:

x^2-x-2=0
(x-2)(x+1)=0
x=2 or x=-1

What we're doing is letting p1 be the statement x^2-x-2=0, p2 be the statement (x-2)(x+1)=0, and q be the statement x=2 or x=-1. It is fair to write p1 <==> p2 <==> q, since each statement implies and is implied by the next.

In both these cases -- doing a proof, and solving an algebraic equation -- we never consider the possibility that the initial statement or, for that matter, the conclusion is false. We simply state them and everybody assumes them to be true.

Once you start asking about "truth value" then you are introducing an additional wrinkle. Rather than asserting that p is true, and p==>q is true and therefore q is true, you are entertaining the possibilities that p and q might be false, and then you are assessing the truth of the overall assertion p==>q.

Going back to the first example, where p is "Joe is happy" and q is "Joe whistles", and we have the compound statement p==>q which means "If Joe is happy then Joe whistles", what if Joe is not happy but he whistles anyway. In other words, what if p is false and q is true? If Joe whistles even when he isn't happy, then the statement "If Joe is happy then Joe whistles" can still be true, so as a matter of convention, we take it to be true. In general, a statement p==>q is taken to be true when p is false. We use this fact in everyday speech: "If Mae West is a good girl, then I'll eat my hat!" is a true statement regardless of my hat consumption, as long as Mae West is not a good girl.

So the truth value of p==>q is generally regarded to be the same as that of not p or q. This true, is reflected in everyday speech. The statement "if you don't eat your dinner then you won't get any dessert" is often rephrased as "you eat your dinner or no dessert!"

The truth value of p<==>q is the same as that of the compound statement p==>q and q==>p. p==>q is false only if p is true and q is false. q==>p is false only if p is false and q is true. The conjunction of p==>q and q==>p is false in either case -- that is if their truth values are different, and true in every other case -- that is if their truth values are the same.

This makes sense, given the meaning of p<==>q, which is that each one implies the other. If the truth values of p and q are the same, then it is true that each implies the other. If the truth values are different, then one of them (the true one) doesn't imply the other one (the false one), so it is not the case the each one implies the other.

Does this answer your question?

--Graeme

Related pages in this website

Methods of Proof goes into the general forms that proofs take (direct, contradiction, infinite descent, etc.)

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