Subsection 1.3.4 Mathematical Claims
Formal logic is the bedrock of mathematics. Mathematicians use statements to represent:
Premises (often called axioms in mathematics): claims that serve as the starting point for building a logically consistent theory that we may hope is useful (or maybe it’s just elegant).
Claims that we would like to try to prove.
Theorems: claims that we can prove must logically follow from the axioms.
Over the centuries, mathematicians have proved many useful theorems.
An example of a theorem that can be proved from the standard axioms of arithmetic:
Theorem 1.3.1.
If \(x\) and \(y \) are even integers, so is \(x + y \text{.}\)
And here’s one that can be proved from the standard axioms of set theory:
Theorem 1.3.2.
The maximum number of elements in the intersection of two sets is the number of elements in the smaller of the two original sets.
There still remain, however, claims that can easily be stated but that have yet to be either proved or disproved.
Conjecture 1.3.3.
Goldbach’s Conjecture remains an unsolved problem: Every even integer greater than 2 is the sum of two prime numbers.
Keep in mind that definitions play a key role in determining the truth of mathematical claims such as these.
Activity 1.3.9.
Consider the claim:
The product of any two primes cannot be prime.
Recall that we considered two possible definitions for prime number: one of them excluded 1; the other didn’t. If we allow 1 as a prime, then our claim is false since \(1 \times 1 = 1\) (which is, by definition, prime). If, however, we don’t allow 1, this claim must be true since any number that is the product of two primes greater than 1 must have both of those numbers as factors (and thus not be prime).