Problem $1.$

Use a truth table to show that there is exactly one assignment of truth values to the letters that

makes the formula

true. Say what this assignment is$.$

$(not{p}^{{?}^{?}}$notq$)harr(pvv(qvvr))$

Problem $2.$

Simplify not$[not[(pvvq{)}^{{?}^{?}}r]vvnotq]$ using fundamental logical equivalences, i$.$e$.$ laws of logic

$($double negation, DeMorgan's, etc.$).$

Problem $3.$

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$).$

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

Explain your reasoning.

Problem $4.$

Let the universe for the variables in the following statement consist of all real numbers. Negate

and simplify AAxAAy$[(|x|=|y|)(y=+-x)]$

Problem $5.$

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.