Lemmas, If and Only If TheoremsReference > Mathematics > Introduction to Proofs
In addition to theorems, we also have things called "lemmas." A lemma is a "little theorem" which is used to help prove a bigger theorem. Usually we call something a lemma if we don't really think we're going to need to re-use it, beyond proving the bigger theorem. Let's look at an example. I'm going to introduce a theorem, and I'll call it "Mr. T's Odd Square Theorem."
Mr. T's Odd Square Theorem: An integer is odd if and only if its square is one more than a multiple of four.
This is something new; we've introduced the terminology "if and only if." This means that the statement works both ways:
If an integer is odd, then its square is one more than a multiple of four.
If an integer's square is one more than a multiple of four, then the number is odd.
In order to prove the theorem, we have to prove both statements above. These are our two lemmas.
Lemma One: If an integer is odd, then its square is one more than a multiple of four.
STATEMENT REASON 1. Let x be any odd number Variable assignment 2. x = 2y + 1, for some integer y Definition of odd number 3. x2 = (2y + 1)2 Squaring property of equality* 4. x2 = 2y(2y + 1) + 1(2y + 1) Distributive property 5. x2 = 4y2 + 2y + 2y + 1 Distributive property 6. x2 = 4y2 + 4y + 1 Combining like terms 7. x2 = 4(y2 + y) + 1 Distributive property 8. ∴ x2 is one more than a multiple of four Definition of multiple
* The squaring property of equality states that the solutions of the new equation contain all solutions of the original equation.
Lemma Two: If an integer's square is one more than a multiple of four, then the number is odd.
This one we will approach as a proof by contradition; we'll assume that it's not true, and arrive at a contradiction.
STATEMENT REASON 1. Assume n is an even number, and n2 = 4k + 1 Assumption 2. n = 2j, for some integer j Definition of even number 3. n2 = 4j2 Squaring property of equality 4. 4j2 = 4k + 1 Transitive property (1, 3) 5. 4j2 - 4k = 4k + 1 - 4k Subtraction property of equality 6. 4(j2 - k) = 1 Combining like terms, Distributive property 7. 1 is a multiple of 4 Definition of multiple
Since 1 is not divisible by 4, this is the contradiction we needed, thus proving Lemma Two.
Since Lemma One and Lemma Two are both true, we have proven both sides of the "if and only if," so we have proved the theorem:
Mr. T's Odd Square Theorem: An integer is odd if and only if its square is one more than a multiple of four
In this section, we introduced you to two different concepts: the idea that "if and only if" indicates that the theorem is "reversible," and a lemma is a mini-proof used to help prove a bigger proof. I've seen lemmas described as "pre-proofs." Something you have to prove before you can prove something else!