Problem $1.$ Which of the following sentences are statements?

a$.$ The moon is made of green cheese.

b$.$ He is certainly a tall man.

c$.$ Two is a prime number.

d$.$ The game will be over by $4$:$00.$

e$.$ Next year, interest rates will rise.

f$.$ Next year, interest rates will fall.

h$.x^2-4=0.$

Problem $2.$ Write the negation of each statement.

a$.$ The processor is fast but the printer is slow.

b$.$ The processor is fast or else the printer is slow.

c$.$ If the processor is fast, then the printer is slow.

d$.$ Either the processor is fast and the printer is slow, or else the file is damaged.

e$.$ If the file is not damaged and the processor is fast, then the printer is slow.

f$.$ The printer is slow only if the file is damaged.

Problem $3.$ Let $A,B,$ and $C,$ and $D$ be the following statements:

A The villain is French.

B The hero is American.

C The heroine is British.

D The movie is good.

Translate the following compound statements into symbolic notation.

a$.$ The hero is American and the movie is good.

b$.$ Although the villain is French, the movie is good.

c$.$ If the movie is good, then either the hero is American or the heroine is British.

d$.$ The hero is not American, but the villain is French.

e$.$ A British heroine is a necessary condition for the movie to be good.

Problem $4.$ Construct truth tables for the following wffs$.$ Note any tautologies or contradictions.

a$.(A^{{?}^{?}}B)vvClongrightarrow{A}^{{?}^{?}}(BvvC)$

b$.A^{{?}^{?}}(A^{\text{'}}vv{B}^{\text{'}}{)}^{\text{'}}$

c$.A^{{?}^{?}}Blongrightarrow{A}^{\text{'}}$

Problem $5.$ Prove the following tautologies by starting with the left side and finding a series of equivalent wffs that will convert the left side into the right side.

a$.(A^{{?}^{?}}{B}^{\text{'}}{)}^{\text{'}}vvBlongleftrightarrow{A}^{\text{'}}vvB$

b$.A^{{?}^{?}}(A^{{?}^{?}}{B}^{\text{'}}{)}^{\text{'}}longleftrightarrow{A}^{{?}^{?}}B$