FOM Book: 4.2

Again, it’s a Friday afternoon, so this is the lightning speed round of some essential 4.2 definitions that struck my fancy so I can be done !

Contrapositive

If you have a statement if p(x), then q(x), the contrapositive is if ~p(x), then ~q(x).

So, say we have the statement if it’s friday, kate is probably crying over all the homework she has for the weekend, the contrapositive would be if it’s not friday, then kate is probably not crying over all the homework she has for the weekend. 

Fun fact! They logically mean the same thing.

If we have an implication statement p(x)⇒q(x), the contrapositive is ~p(x)⇒~q(x). So if we have an implication statement like kate drinking coffee in class implies a snarky kate, the contrapositive of this is kate not drinking coffee in class implies a non-snarky kate. 

Theorem 4.13: ~[p(x) and q(x)] is logically equivalent to “~p(x) or ~q(x)”.

Theorem 4.14: ~[p(x) or q(x)] is logically equivalent to “~p(x) and ~q(x)”.

Converse

If you have a statement if p(x), then q(x), the converse is if q(x), then p(x).

So, say we have the statement if it’s friday, kate is probably crying over all the homework she has for the weekend, the converse would be if kate is crying over all the homework she has for the weekend, then it’s friday.

If we have an implication statement p(x)⇒q(x), the converse is q(x)⇒p(x). So if we have an implication statement like kate drinking coffee in class implies a snarky kate, the converse of this is a snarky kate implies a kate drinking coffee in class.

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