Week 2: Quantifiers, Implication

This week, I learned about quantifiers, implication and verifying implication. This part wasn't completely surprising since it is the same contents as current Calculus class. It was helpful to have same part in two different classes at the same time. It gave me a better understanding of material.

When verify quantified claims, if it is an universal claim, show there is no 'counter-example'. In other words, if there is at least one counter-example, an universal claim becomes false. But if it is an existential claim, show there is at least one example then it is true. When there is no example, it is a false claim.

Implication is 'If P, then Q.' P is called the antecedent and Q is called the consequent. The converse of implication is 'If Q, then P.' and there is no connection between the implication and the converse. Its negation is 'If P, then not Q.' and contrapositive is 'If not Q, then not P.'

Implication can be verified in a similar manner with verifying quantified claims. Following us 'Truth Table'.

P	Q	P -> Q
T	T	T
T	F	F
F	T	T
F	F	T

When the antecedent is false, the whole implication is true. This is because 'P' is false so it is an empty set. If it is an empty set there is no counter-example can be found. Therefore, the implication is true. When I first read this part, it was hard to accept this. What I thought is that implication should be false because 'P' is false so whole statement cannot be established. It is quite still uncomfortable but at least I understood why the implication is true when 'P' is false.