Problem $1.$ Justify each step in the proof sequence of $A^{\text{'}}{?}^{{?}^{?}}{B}^{{?}^{?}}[Blongrightarrow(AvvC)]longrightarrowC$

$A^{\text{'}}$

$B$

Blongrightarrow$(AvvC)$

AvvC

$(A^{\text{'}}{)}^{\text{'}}vvC$

$A^{\text{'}}C$

$C$

Problem $2.$ Prove that the argument is valid.

a$.(Avv{B}^{\text{'}}{)}^{\text{'}}{?}^{{?}^{?}}(BlongrightarrowC)longrightarrow(A^{\text{'}}{?}^{{?}^{?}}C)$

b$.(A^{\text{'}}longrightarrow{B}^{\text{'}}{)}^{{?}^{?}}{B}^{{?}^{?}}(AlongrightarrowC)longrightarrowC$

c$.[Alongrightarrow(BlongrightarrowC){]}^{{?}^{?}}(Avv{D}^{\text{'}}{)}^{{?}^{?}}$Blongrightarrow$(DlongrightarrowC)$

Problem $3.$ Prove that the following argument of the defense attorney is valid.

If my client is guilty, then the knife was in the drawer. Either the knife was not in the drawer, or Jason Pritchard saw the knife. If the knife was not there on October $10,$ it follows that Jason Pritchard did not see the knife. Furthermore, if the knife was there on October $10,$ then the knife was in the drawer and also the hammer was in the barn. But we all know that the hammer was not in the barn. Therefore, ladies and gentlemen of the jury, my client is innocent.

Problem $4.$ What is the truth value of each of the following wffs in the interpretation where the domain consists of the integers, $A(x)$ is $"x<mn\; id="EditorElement\_1202541">5</mn><mtext\; id="EditorElement\_1202527"></mtext><mi\; id="EditorElement\_1202528">"</mi>$ and $B(x)$ is

a$.(EEx)A(x)$

b$.$ AAx$[A(x)longrightarrowB(x)]$

c$.$ EEx$[A(x{)}^{{?}^{?}}B(x)]$

Problem $5.$ Using the predicate symbols shown and appropriate quantifiers, write each English language statement as a predicate wff$.($The domain is the whole world.$)B(x)$:$x$ is a ball; $R(x)$:$x$ is round; $S(x)$:$x$ is a soccer ball

a$.$ All balls are round.

b$.$ Not all balls are soccer balls.

c$.$ All soccer balls are round.

d$.$ Some balls are not round.

e$.$ Some balls are round but soccer balls are not.

Problem $6.$ Three forms of negation are given for each statement. Which is correct?

a$.$ Nobody is perfect.

Everyone is imperfect.

Everyone is perfect.

Someone is perfect.

b$.$ All swimmers are tall.

Some swimmer is not tall.

There are no tall swimmers.

Every swimmer is short.