What is the mathematical definition for double implication?

Answer 1

In order for you to read mathematical text accurately, you must know what we mean when we see or write factual implications. Compare the two sentences from ordinary talk:

. . . . . 1. If it stops raining, [then] I'll go to the market.

. . . . . 2. If I win the lottery, [then] I'll buy a new car.

The two sentences have parallel structure (If A, then B), but the usual meaning is quite different. In #1, the inference one picks up is that if it does not stop raining, I won't go to the market. In #2, on the other hand, the failure to win the lottery is hardly grounds for keeping the old car; I might decide to buy a new car on credit, for instance.

Many assertions in math books are of the form "If A, then B". Mathematics is hard enough without ambiguous statements! Which of the two statements above is giving the correct, agreed mathematical usage?It's #2. We'll come back to these examples later.

We often write A ==> B for "if A, then B". This means only that A implies B (B is a consequence of A, as the arrow suggests). Use common sense to see that this is equivalent to its contrapositive, not B ==> not A (for if you did get A, ...).

. . . . . Statement: If two numbers are odd, then their sum is even.

. . . . . Contrapositive: If the sum of two numbers is odd, then at least one of them must be even.

This is logically different from the inverse, not A ==> not B, and its contrapositive, B ==> A (the converse of A ==> B). Continuing the above example:

. . . . . Inverse: If (at least) one of two numbers is even, then their sum is odd

which you can see is false.

The inverse (and converse) are saying that you can't have B without A, which can be rephrased (even in common parlance) as B only if A:

. . . . . You'll get the flu only if you are exposed to an influenza virus.

The converse of this statement is false; otherwise we'd all be ill! In particular, the correct mathematical rendition of the connoted meaning of #1 is:

(*). . . . . I will go to the market if and only if it stops raining.

When we state an implication in mathematics, it contains no information about the converse (or equivalently, the inverse) statement.

Combining the two implications, A ==> B and B ==> A, we write B <==> A or A <==> B (B if and only if A); the assertions A and B carry the same actual information, i.e., are equivalent.

In mathematics, it was decided that "or" is non-exclusive. That is, when we say that A or B holds, we mean that at least one (so possibly both A and B -- we do not exclude that) of the two statements are true.

A counterexample to an assertion is an example that shows that the assertion is false. For instance, here's a stupid statement to ponder:

. . . . ."Every real number is a rational number (usual fraction a/b, with a and b integers, and bnon-zero)."

Come on! you might say. \pi is not rational! That is correct: \pi is a counterexample to (example against) the assertion.

In mathematics, statements are to be proved true, or shown to be false by finding a counterexample. That's right, just one. But often enough, once you've found one counterexample, there are many more.

For emphasis, I remind you that no incomplete list of examples suffices, of itself, to prove that a mathematical statement is true; it only says that you haven't yet found a counterexample!

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The following questions will be part of the first homework assignment, and they are to be submitted as WIproblems.

(There is no reason why you should be able to do these problems quickly, but you can do them if you persist.)

. . . . . 1. The context here is pairs of whole numbers [integers]. In each case,

. . . a) determine whether A ==> B, and whether B ==> A.

. . . b) write the outcome in one sentence, using one of the following: if, only if, if and only if, neither implies the other, as in (*).

i) A: Both numbers are even numbers. . . . B: Their sum is even.

ii) A: Both numbers are odd numbers. . . . B: Their product is odd.

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. . . . . 2. Carry out (a) and (b) in #1 for the following assertions about functions f(x):

i) A: f(x) is a polynomial function; . . . B: f'(x) (the derivative of f(x)) is a polynomial function.

ii) A: f(x) is a rational function; . . . B: f'(x)(the derivative of f(x)) is a rational function.

(A rational function is a function that can be written in the form f(x) = p(x)/q(x), where p(x) and q(x) are polynomials. That is how one writes the general rational function.

You may freely use the algebraic facts about polynomials.)

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Steven Zucker