Difficulty in understanding negation and parentheses in math logic

Question

I'm working on a exercise from "Logic for Mathematicians" by A.G.Hamilton.I'm taking a logic and set theory course and I have lost my first year "foundation of math" goods. I'm faced with this proposition : ¬((p→(¬q))→r) I reviewed my past material and I couldn't figure it out how should I deal with the above negation and parentheses. Is "negation" for the whole "((p→(¬q))→r)" ? how should I read the above proposition? is it "if p→(¬q)) then r" or is it "if p then (¬q))→r" ? How this negation and parentheses workout together? I'm ashamed that I have forgotten a lot of this basic knowledge. FINAL NOTE: This is not a homework question. Thanks for your time and help in advance!

Answer 1

You should remember that $a→b$ is equivalent to $¬a∨b$, you can prove by truth table.Therefore,for $¬((p→(¬q))→r)$ ,we can translate it to ¬(if(if p then ¬q) then r) or

=$¬((¬p∨¬q)→r)$

=$¬((p∧q)∨r)$

=$((¬p∨¬q)∧¬r)$ (De Morgan's laws)

=$((p→¬q)∧¬r)$

(if p then ¬q) and ¬r

How to read parentheses and negation

((¬p)∧r)=(¬p∧r)=¬p∧r

¬((p)∧r)=¬(p∧r)=¬p∨¬r