Examples of Basic Proofs


Proving "If A, then B" directly

Start by assuming that A is true, and logically argue that B must follow.

Proof:
Suppose A.
Then … (( insert sequence of logical arguments here; probably will involve the definitions of the objects involved ))
Therefore, B.   ∗


Proving "If A, then B" by contrapositive

This is equivalent to proving "If (Not A), then (Not B)" directly.

Proof:
To prove that "If A, then B," we will prove the equivalent statement "If (Not B), then (Not A)."
Suppose Not B.
Then … (( insert sequence of logical arguments here; probably will involve the definitions of the objects involved ))
Therefore, Not A.   ∗


Proving "If A, then B" by contradiction

Given the assumptions in A, show that B must be true because it cannot possibly be false.

Proof:
Suppose A.
Seeking a contradiction, suppose Not B.
Then … (( make logical conclusions until you come to two statements that contradict each other, such as "X is true" and X is false" ))
But this is a contradiction because … (( explicitly mention the contradictory statements ))
Therefore our initial assumption that "Not B" cannot possibly be true (i.e. "Not B" must be false).
Thus B must be true.   ∗


Disproving an implication "If A, then B" with a counterexample

To show such an implication is false you just need to exhibit one counterexample, i.e. one example where A is true but B is false.

Counterexample:
______ is a counterexample
because … (( state how A is true in your example ))
but … (( state how B is false in your example )).   ∗