CSC165 - Week four - Bi-implications, Transitivity, Mixed Quantifiers, Proof

It's the fourth week into my university journey, and 1/3 into the semester, who would have believed that?
In this week's lecture, we learned bi-implication, transitivity, mixed quantifier and proof.
Bi implication is just a two-way if...then statement.

P(x) <==> Q(x)

 We turned it into a conjunction of two disjunctions

not P(x) or Q(x) and not Q(x) or P(x)

, and a disjunction of two conjunctions.

not P(x) and not Q(x) or P(x) and Q(x)

We looked at transitivity, which means when p is related to q, and q is related to r, then p is related to r. We learned that transitivity is always true by reducing its negation to a contradiction.

((P=>Q)^(Q=>R)) => (P=>R)

neg= (P=>Q) and (Q=>R) or not(P=>R)

=(not P or Q) and (not Q or R) or (P and R)

which is a contradiction

Next up we talked about mixed quantifiers, and how when we mix two different quantifiers, the statement's meaning is twisted when we change the order of the quantifiers. 

for example:

let x,y be real numbers,

1) for all x, there exists a y, y>x

2) there exists a y, for all x, y>x

these two sentences have the same quantifiers, but because we

changed the order of the quantifiers, they mean different things.

1) for every real number, there is a bigger number

2) there is a real number greater than all other real numbers

obviously we can see that statement 1 is true and statement 2 is false,

the only difference is the order of the quantifiers

We finished the lecture by talking about proof, proof is an extremely important concept in science, nothing is to be believed until its proven.

a proof can be worked top-down or bottom-up, and if we can connect some statement like linking some nodes and connect top and bottom, we can prove it.