Lecture 6: inverses

Today's was a definition heavy lecture. We played with left-, right-, and two-sided inverses. I also discussed some important meta points about "for all" and "there exists".


Important note about definitions: When we give a definition, we usually say something like "Definition: X is Y if Z". This is sometimes confusing shorthand, because what we really mean is "the definition of X being Y is Z". In particular, you should read that "if" as an "if and only if" (but only in the case of definitions).

Some (unproven but useful) facts about inverses

Note: feel free to use these facts on the homework, even though we won't have proved them all.

We'll probably prove one of these tomorrow, the rest are similar.

A quick proof using inverses

Here is a shorter proof of one of last week's homework problems that uses inverses:

Claim: If A∣ ≥ ∣B then B∣ ≤ ∣A.

Proof: Suppose A∣ ≥ ∣B. By definition, that means there is some function f: AB that is onto. Since f is onto, it has a right inverse g. By definition, this means that f ∘ g = idB. Notice that this is the same as saying the f is a left inverse of g. Therefore g has a left inverse, and so g must be one-to-one. Therefore, since there exists a one-to-one function from B to A, B∣ ≤ ∣A.

Compare this to the proof in the solutions: that proof requires us to come up with a function and prove that it is one-to-one, which is more work.

Meta-notes about proving things (negating quantifiers)

Here I add a bit more detail to an important point I made as an aside in lecture.

Suppose P(x) is a statement that depends on x. For example, P(x) might be "x has purple hair" or "x is a piece of chalk" or "for all y ∈ N, if f(y) = x then y = 7". These statements are called "predicates"

To prove a statement of the form "for all x ∈ A, P(x)", you must consider every possible value of x. To disprove such a statement, you only need to find one x for which P(x) does not hold.

For example, the definition of one-to-one says that "for all x and y, if f(x) = f(y) then x = y". To prove that a function is one-to-one, you must either consider every possible element of the domain, or give me a general argument that works for any element of the domain.

However, to prove that a function is not one-to-one, you only need to find one pair of elements x and y with x ≠ y but f(x) = f(y).

Summarizing this:

"not (for all x, P(x))" is equivalent to "there exists x such that not P(x)".

Similarly, to prove a statement of the form "there exists x such that P(x)", it suffices to give me a single example of an x having property P. To disprove such a statement, you must consider all possible counterexamples.

Summarizing this:

"not (there exists x such that P(x)) is equivalent to "for all x, not P(x)"