$($a$)$ Let A$,$B$,$C denote languages. Notation $<=$P means polynomial time reducibility.

$$ Statement: If A $<=$P B and B in NP$,$ then A in NP$.$

Is the statement true?

Circle the answer: yes no

$$ Statement: If A in P and B in NP and B $=,\backslash$Sigma $$ then A $<=$P B$.$

Is the statement true?

Circle the answer: yes no

$$ Statement: If A is NP$-$complete and A $<=$P B then B is NP$-$hard.

Is the statement true?

Circle the answer: yes no

$$ Statement: If A in NP and A $<=$P B then B in NP$.$

Is the statement true?

Circle the answer: yes no

$($b$)$ This is about Boolean formulas and their satisfiability.

Statement: A formula in $3$ conjunctive normal form that consists of three clauses has exactly six occurrences of literals, including repetitions.

Is the statement true?

Circle the answer: yes no

$($c$)$ Let V $=\{$p$1,$p$2,$p$3,$p$4,$p$5\}$ be a set of Boolean variables. Consider a Boolean formula:

$$ Statement: Formula $\backslash$phi is in conjunctive normal form.

Is the statement true?

Circle the answer: yes no

$$ Statement: Formula $\backslash$phi is in $3$ conjunctive normal form. Is the statement true?

Circle the answer: yes no

$$ Statement: If $\backslash$phi is satisfied by an assignment of logical values to the variables in V $,$ then all but one variables in V are set to false. Is the statement true?

Circle the answer: yes no

$$ Statement: Formula $\backslash$phi is satisfied by precisely five different assignments of logical values to the variables in V $.$

Is the statement true?

Circle the answer: yes no