How do you know it is true?

A picture suggests. A proof explains.

A definition fixes what a word means. An integer $n$ is even if it can be written as $n=2k$ for some integer $k$. A claim states something that may be true or false. A theorem is a proved mathematical result; propositions and lemmas are other names used to organize results. A corollary follows from an earlier result.

An example helps you understand a claim. It does not prove a statement about every possible input.

Try to break a claim

Consider “For every real number $x$, $x^2\ge x$.” It works at $x=2$, $x=3$, and $x=-1$. Is that enough?

At $x=1/2$, the square is $1/4<1/2$. One counterexample disproves a universal claim. The experiment visualizes why: between zero and one, the square curve lies below the straight line. Testing many successful cases would never make that one failure disappear.

Follow a direct proof

Claim: The sum of two even integers is even.

1. Let $a$ and $b$ be any even integers. By definition, $a=2r$ and $b=2s$ for integers $r,s$.
2. Add: $a+b=2r+2s=2(r+s)$.
3. Since $r+s$ is an integer, the result has the form “2 times an integer.” It is even by definition.

The proof covers all even inputs because $r$ and $s$ were arbitrary. Checking $2+4=6$ would cover only one pair.

Which direction does an implication run?

Let A mean “n is divisible by 4” and B mean “n is even.” Then $A\Rightarrow B$.

• A is sufficient for B: divisibility by 4 guarantees evenness.
• B is necessary for A: divisibility by 4 requires evenness.
• B is not sufficient for A: 6 is even but not divisible by 4.

An “if and only if” statement requires both directions. For integers, “n is even if and only if n² is even” makes two promises, not one.

A proof by contradiction

To prove a claim by contradiction, assume it fails and show that the assumption forces an impossibility.

Claim: There is no largest integer. Suppose there were a largest integer $N$. But $N+1$ is also an integer and is larger than $N$. That contradicts the meaning of “largest.” Therefore no largest integer exists.

Keep the assumptions visible. For example, $x^2=1\Rightarrow x=1$ is false over the reals because $x=-1$ is possible. It becomes true if we add the assumption $x\ge0$.

Ready for the source

You now have a way to approach unfamiliar notation: draw a small case, calculate an example, test the assumptions, then read the general statement. Module 1 connects these habits to the original lecture notes.