Problem $1.$

Verify that $[p(qr)][(pq)(pr)]$ is a tautology.

Problem $2.$

Assuming proposition $q$ is true $($T$),$ determine all truth value assignments for the propositions $p,r,$ and $s$ for which the truth value of the following compound proposition is false $($F$).$

Explain your reasoning.

$(q[(not{p}^{{?}^{?}}r)vvnots])vv[s(not{r}^{{?}^{?}}q)]$

Problem $3.$

Consider each of the following arguments. If the argument is valid, identify the rule of inference that establishes its validity. If not, explain why.

a$)$ Andrea can program in $C++,$ and she can program in Java. Therefore, she can program in C$++.$

b$)$ If Ron's computer program is correct, then he'll be able to complete his computer science assignment in at most two hours. It takes Ron over two hours to complete his computer science assignment. Therefore, Ron's computer program is not correct.

c$)$ If interest rates fall, then the stock market will rise. Interest rates are not falling. Therefore, the stock market will not rise.

Problem $4.$

Let the universe for the variables in the following statement consist of all real numbers. Negate and simplify $)]=(|y|)=[(+-x.$

Problem $5.$

A perfect square is an integer which is also square of an integer. More formally, ninZ is a perfect square if $n=a^2$ for some ainZ.

Prove or disprove the follo