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Thursday, December 5, 2013

Bijections and Finite Sets: Proof by If And Only If

Here's a proof that requires a bit of subtlety...

Prove that a nonempty set T_{1} is finite if and only if there is a bijection from  T_{1} onto a finite set  T_{2}.

We can use the following:

  • Definition     (a)  The empty set is said to have zero elements.
  • (b)  If n\in \mathbb{N},  a set S is said to have n elements if there exists a bijection from the set         \mathbb{N}_{n} := {1, 2, ..., n} onto S.
  • (c)  A set S is said to be finite if it is either empty or has n elements for some n\in \mathbb{N}.
  • (d) A set S is said to be infinite if it is not finite.
Step 1: If
Assume there is a bijection from T_{1} onto T_{2}. We're given that T_{2} is finite. We know that by definition, every element of one set is paired with exactly one element of the other set, and every element of the other set is paired with exactly one element of the first set (T_{1}). Since T_{2} is finite, T_{1} is finite.

Step 2: Only If
Assume T_{1} is finite set, which is nonempty as described in the original problem statement. This means that by the definition above, T_{1} has n elements for some n\in \mathbb{N} and there exists a bijection from the set \mathbb{N}_{n} := {1, 2, ..., n} onto T_{1}. Let's call that bijection f.

We also are given that T_{2} is a finite set. Assuming that T_{2} is not empty, we know by the definition above that there exists a bijection from the set \mathbb{N}_{n} := {1, 2, ..., n} onto T_{2}. Let's call that bijection g.

The composition g\circ f would map T_{1} into T_{2}, and since the composition of two bijections is a bijection, g\circ f is a bijection from T_{1} onto T_{2}.

QED 

Tuesday, December 3, 2013

Exponents and Factorials: Mathematical Induction Proof

Easy one here...

Prove that 2^{n}< n! for all n\geq 4, n\in \mathbb{N}.

Step 1: Show P(4) is true.
Our base case is k = 4. 2^{4}<4! \Rightarrow 16 < 24. OK.

Step 2: Show that (if we assume P(k) is true), P(k+1) is true. (Induction hypothesis). This means we should have 2^{k+1}<(k+1)!.

2^{k+1}<(k+1)!
can be rewritten as
2\cdot2^{k}<(k+1) k!

For k\geq 4, 2 < k + 1, and by the induction hypothesis, we assume 2^{k}< k!, so we can see that
2^{n}< n!
for all n\geq 4, n\in \mathbb{N}

QED



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