a systemic approach to take a set of true facts to reach a conclusion
a statement must contain a property and an object
“If A, then B”
Let A be True
Then [apply A.property to A.object]
Consider B.object
[Get to B.property in some creative way]
$\therefore$ B is True $\square$
For Example: " if $x$ if even, then $x^2$ is even"
A.object = x
A.property = even
B.object =$x^2$
B.property = even
Let $x \in \mathbb{R}$ be even
Then $x = 2k, k \in \mathbb{Z}$
Consider $x^2$
$x^2 = (2k)^2 = 4k^2 = 2(2k^2)$
Since $2, k \in \mathbb{Z}$, Let $2k^2 = q, q \in \mathbb{Z}$
$ \Rightarrow x^2 = 2q$ for some 2
$ \therefore $ we have shown that if x is even, then $x^2$ is even$
This type of proof is called a direct proof because it logically arrives at a conclusion starting from the assumed facts. you would usually use this when A is simple a can easily sub into B.
But there is also indirect proofs which prove a statement by proving either the opposite input leads to the opposite result (if not A, then not B) or the opposite input leads to a contradiction, forcing the the normal input to make sense. you would use them typically when A is more complicated because they allow us to work with B instead.
“If A, then B”
We will prove using the contrapositive (if $\neq A$, then $\neq B$)
Let $\neg$ B.object
Then $\neg$ B.property
Consider $\neg$ A.object
[Get to $\neg$ A.property in some creative way]
$\therefore \neg$ A
So we have shown A implies B by contrapositive
“If A, then B”
Let A be True
Then [apply A.property to A.object]
Assume $ \neg$ B
Consider A.object
[do some math until something breaks horribly (e.g. 1 = 2)]
Thus our assumption was False, so B must be True
$\therefore$ We have shown A implies B by contradiction $\square$
if a statement uses “$\Leftrightarrow$” or “iff (if and only if)” that means both objects imply each other. like for $ A \Leftrightarrow B$ means $ A \Rightarrow B$ and $ B \Rightarrow A$, so you have to prove both statements