1- Determine the truth value of the following statement if the domain of each variable consists of all integers ? [0.25pt] x: x2 x [T / F]

2- P (x) is the statement x2 < 8 and the domain consists of the positive integers less than 6. Write out the proposition xP(x) using disjunctions, conjunctions, and negations (i.e. rewrite the propositions without using quantifiers). [0.5pt]

a) (P(1) P(2) P(3) P(4) P(5))

b) (P(1) P(2) P(3) P(4) P(5) P(6))

c) (P(1) P(2) P(3) P(4) P(5) P(6))

d) (P(1) P(2) P(3) P(4) P(5) P(6) P(7) )

3- P (x) is the statement x2 < 8 and the domain consists of the positive integers less than 6. Write out the proposition xP(x) using disjunctions, conjunctions, and negations (i.e. rewrite the propositions without using quantifiers). [0.5pt]

a) P(1) P(2) P(3) P(4) P(5) P(6) P(7)

b) P(1) P(2) P(3) P(4) P(5)

c) P(1) P(2) P(3) P(4) P(5) P(6) P(7)

d) (P(1) P(2) P(3) P(4) P(5) )

4- Construct a combinatorial circuit using inverters, OR gates, and AND gates that produce the output ((p q r ) (p q r ) ) from input bits p and q. [