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Proofs
Always just assume the conditional part of the statement
Funny Spec words
This is in "the realm of logic"
- Negation: The opposite of the statement, i.e. asserts the opposite
- e.g. If: x is equal to 6
- Negation: x is not equal to 6
- e.g. If: n is greater than 12
- Negation: n is less than or equal to 12
- We need negations to construct a contrapositive!
- Converse: If, then content of conditional statement switched
- e.g. If a polygon is a quadrilateral then it has exactly four sides.
- Converse: If a polygon has exactly four sides then it is a quadrilateral.
- e.g. If you were born in Australian, then you have been in Australia
- Converse: If you have been in Australia, then you were born in Australia.
- If the converse of a true statement is also true, then
- $A \leftrightarrow B$
- Inverse: negate each of the if and then content
- e.g. If a polygon is a quadrilateral then it has exactly four sides
- Inverse: If a polygon is not a quadrilateral then it does not have exactly four sides.
- Contrapositive: Converse + Inverse, i.e. swap if, then content, and also negate!
- i.e. If a polygon does not have exactly four sides, then it is not a quadrilateral
- De Morgan's Law
- not(P and Q) = not(P) or not(Q)
- not(P or Q) = not(P) and not(Q)
Summary
Given a true conditional statement:
$$If \ A \ then \ B
$$
- The converse (If $B$ then $A$) may or may not be true.
- The inverse (If not $A$ then not $B$) may or may not be true.
- The negation (If $A$ then not $B$) must be false given the statement is true. Else, it may or may not be true.
- The contrapositive (If not $B$ then not $A$) has the same truth value as the original statement.
Proof by Contrapositive
Example: "If $n^{2}+ 4n + 1$ is even, then $n$ is odd."
What to do? Find the contrapositive
Proof:
Contrapositive: "If n is even, then $n^{2}+ 4n + 1$ is odd."
let $n$ be even, i.e. $n=2k$ where $k \in \mathbb{Z}$
$\therefore n^{2}+ 4n + 1 = (2k)^{2}+ 4(2k) + 1$
$= 4k^{2}+ 8k + 1$
Hence $n^{2}+ 4n + 1 = 2(2k^{2}+4k) + 1,k \in \mathbb{Z},2k^{2}+4k \in \mathbb{Z}$
Thus $n^{2}+ 4n + 1$ is odd since $2k^{2}+4k$ is an integer.
The contrapositive is true, hence the statement is true
