Proofs and Truth Tables

[L3]

I'm not sure if this is the right place to ask this question, but the site I'm using puts the lesson under geometry, so I'll ask the question under geometry.

Frankly I'm very very baffled. And I have a sneaking suspicion I'm missing something but I'm not sure what it is.

So, as far as I can figure out, and also the fact that the site gives it, the truth table is,

A B Implication

T T T
T F F
F T T
F F T

T being for True F for false.

And the contrapositive takes a sentance like 'if A then B' and turns it in to 'if not B then not A' and the two sentances are interchangeable.

Okay, and I was fine with all that until I got to the lesson not (A and B) being equivalent to (not A) or (not B). And the site gives a truth table.

A | B | A and B | not(A and B) | not A | Not B | (not A) or (not B)

T | T | T | F | F | F | F
T | F | F | T | F | T | T
T | F | F | T | F | T | T
F | F | F | T | T | T | T



Which I just don't understand, and I have a feeling it's because I'm missing some sort of key concept here, but I don't know what.

I mean, the first table said that the only time the Implication is false is when A is true and B is false. So why in the last three lines, when not A is false and Not B is false, is (not A) or (not B) false? Why isn't it true?

I'm sorry if I'm being very slow about this, I'm find this whole thing quite complicated and difficult.

 

[soroban]

Hello, L3!

In proving one of DeMorgan's Laws, we are working with an equivalance.

We want to prove that: .\(\displaystyle \sim(A \wedge B) \:\Longleftrightarrow\\sim A) \vee (\sim B)\)

Look at their truth table again.
The final columns look like this:

. . \(\displaystyle \begin{array}{c|c|c} \sim(A \wedge B) & \Longleftrightarrow & (\sim A) \vee (\sim B) \\ \hline F && F \\ T && T \\ T && T \\ T&&T \end{array}\)
. \(\displaystyle \text{and in the }\overbrace{\text{final column}}^{\uparrow}\text{, you get "all T's".}\)

 

[L3]

Ah, thank-you muchiles! I'm finding this a rather difficult math lesson .