hi,

I'm embarking on a course on Coursera in Introduction to Mathematical Thinking. I seem to be going seriously wrong somewhere because none of my answers are coming back as being correctly answered.

Q1 - . Build a truth table to prove the claim I made earlier that φ ⇔ ψ is true if φ and ψ are both true or both false, and φ ⇔ ψ is false if exactly one of φ, ψ is true and the other false. (To constitute a proof, your table should have columns that show how the entries for φ ⇔ ψ are derived, one operator at a time.)

http://[Q1](https://i.imgur.com/WLi5xE8.jpg)

Questions: If we can assume that F(alse) = 0, while T(rue) = 1 and equivalence is multiplying the two together then we'll get:

T F => F

T T => T

F T => F

F F => T

but that means that 0 x 0 should equal 1, which it obviously doesn't yet I've heard consistently that F F => T -- can someone clarify where I'm going wrong or where I have misunderstood?

Q2 -Build a truth table to show that (φ ⇒ ψ) ⇔ (¬φ ∨ ψ) is true for all truth values of φ and ψ. A statement whose truth values are all T is called a logical validity, or sometimes a tautology.

http://[Q2](https://i.imgur.com/tnmvIPC.jpg)
I'm not sure what went wrong?!to

Questions: Is there an easy way to ensure that I've included all the correct notations in the truth table?

Q3 - Build a truth table to show that (φ 6⇒ ψ) ⇔ (φ ∧ ¬ψ) is a tautology.

http://[Q3](https://i.imgur.com/oJf7wkL.jpg)
Again I'm not sure where I'm going wrong here!

Jumping to 9) to stay on the same theme

http://[Q4](https://i.imgur.com/QrwWU0k.jpg)
I must be misunderstanding something here to keep getting these so wrong, any assistance would be much appreciated.