# 8.E: Chapter Seven (Exercises)

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##### Exercise $$\PageIndex{1}$$: Identifying Instances of the Rules

Identify which (single) rule is being exemplified in each inference. Remember that the underline separates premises from their conclusion.

A. \begin{align*} & [(A \wedge \neg B) \rightarrow (P \vee Q)] \\ & \overline{[ \neg (P \vee Q) \rightarrow \neg (A \wedge \neg B)]} \end{align*}

B. \begin{align*} & [(P \vee Q) \vee \neg (A \wedge \neg B)] \\ & \underline{\neg \neg (A \wedge \neg B) \ \ } \\ & (P \vee Q) \end{align*}

C. \begin{align*} & \underline{[((Z \vee X) \rightarrow Y) \wedge (Y \leftrightarrow W)]} \\ & ((Z \vee X) \rightarrow Y) \end{align*}

D. \begin{align*} & [((Z \vee X) \rightarrow Y) \rightarrow (Y \leftrightarrow W)] \\ & \underline{((Z \vee X) \rightarrow Y) \ } \\ & (Y \leftrightarrow W) \end{align*}

E. \begin{align*} & [ \neg B \rightarrow (P \vee Q)] \\ & \underline{\neg (P \vee Q) \ \ } \\ & \neg \neg B \end{align*}

F. \begin{align*} & [((Z \vee X) \rightarrow Y) \vee (Y \leftrightarrow W)] \\ & \underline{[(((Z \vee X) \rightarrow Y) \rightarrow T) \wedge ((Y \leftrightarrow W) \rightarrow \neg (W \leftrightarrow Z))]} \\ & [T \vee \neg (W \leftrightarrow Z)] \end{align*}

G. \begin{align*} & \underline{(((Z \vee X) \vee Y) \vee T)} \\ & (Z \vee (X \vee Y)) \vee T) \end{align*}

H. \begin{align*} & \underline{[ \neg (P \vee Q) \vee (A \leftrightarrow \neg B)]} \\ & [(P \vee Q) \rightarrow (A \leftrightarrow \neg B)] \end{align*}

I. \begin{align*} & \underline{[(P \vee Q) \rightarrow (A \leftrightarrow \neg B)]} \\ & [ \neg (P \vee Q) \vee (A \leftrightarrow \neg B)] \end{align*}

J. \begin{align*} & \underline{(((Z \vee X) \rightarrow Y) \rightarrow T) \ } \\ & (((Z \vee X) \rightarrow Y) \rightarrow T) \vee [((Y \leftrightarrow W) \rightarrow \neg (W \leftrightarrow Z))] \end{align*}

K. \begin{align*} & [((Y \leftrightarrow W) \rightarrow \neg (W \leftrightarrow Z)) \rightarrow (T \vee \neg (W \leftrightarrow Z))] \\ & \underline{((Y \leftrightarrow W) \rightarrow \neg (W \leftrightarrow Z)) \ } \\ & (T \vee \neg (W \leftrightarrow Z)) \end{align*}

L. \begin{align*} & (((Z \vee X) \rightarrow Y) \rightarrow T) \\ & (T \rightarrow \neg (W \leftrightarrow Z)) \\ & \overline{[((Z \vee X) \rightarrow Y) \rightarrow \neg (W \leftrightarrow Z)]} \end{align*}

##### Exercise $$\PageIndex{2}$$: Using the First 4 Rules

Derive the conclusion (after the slash) from the numbered premises using only the first 4 rules we learned in section 7.2.

A.

• 1. ~C$$\supset$$ (A$$\supset$$C)
• 2. ~C /~A

B.

• 1. ~W$$\supset$$ [~W$$\supset$$ (X$$\supset$$W)]
• 2. ~W / ~X

C.

• 1. (P  $$\rightarrow$$  Q)
• 2. (Q $$\rightarrow$$ R)
• 3. (R $$\rightarrow$$ S)
• 4. (S $$\rightarrow$$ T) / (P $$\rightarrow$$ T)

D.

• 1. (((Z $$\vee$$ X) $$\rightarrow$$ Y) $$\rightarrow$$ T)
• 2. (T $$\rightarrow$$ $$\neg$$ (W $$\leftrightarrow$$ Z))
• 3. ((Z $$\vee$$ X) $$\rightarrow$$ W)
• 4. (W $$\rightarrow$$ Y) / $$\neg$$ (W $$\leftrightarrow$$ Z

E.

• 1. ~M $$\vee$$ (B $$\vee$$ ~T)
• 2. (B $$\supset$$ W)
• 3. ~~M
• 4. ~W / ~T

F.

• 1. (~S $$\supset$$ D)
• 2. [~S ∨ (~D $$\supset$$ K)]
• 3. ~D / K

G.

• 1. [A $$\supset$$ (E $$\supset$$ ~F)]
• 2. [H $$\vee$$ (~F $$\supset$$ M)]
• 3. A
• 4. ~H / (E $$\supset$$ M)

H.

• 1. [G $$\supset$$ [~O $$\supset$$ (G $$\supset$$ D)]]
• 2. (O $$\vee$$ G)
• 3. ~O / D

I.

• 1. P$$\supset$$ (G$$\supset$$T)
• 2. Q$$\supset$$ (T$$\supset$$E)
• 3. P
• 4. Q / (G$$\supset$$E)

J.

• 1. (~S $$\supset$$ D)
• 2. ~S $$\vee$$ (~D$$\supset$$ K)
• 3. ~D /K

K.

• 1. X $$\rightarrow$$ (Y $$\rightarrow$$ Z)
• 2. X $$\rightarrow$$ (Z $$\rightarrow$$ W)
• 3. (T $$\vee$$ X)
• 4. ~T / (Y $$\rightarrow$$ W)

L.

• 1. X $$\vee$$ (Y $$\wedge$$ Z)
• 2. (Y $$\wedge$$ Z) $$\rightarrow$$ W
• 3. (X $$\rightarrow$$ T)
• 4. $$\neg$$ T /W
##### Exercise $$\PageIndex{3}$$: Using the First 8 Rules

Derive the conclusion (after the slash) from the numbered premises using only the first 8 rules (the rules of implication) we learned in sections 7.2 and 7.3.

A.

• 1. (~A $$\rightarrow$$ H)
• 2. (R $$\rightarrow$$ ~B)
• 3. (~A $$\vee$$ R) / (H $$\vee$$ ~B)

B.

• 1. E $$\supset$$ (A $$\bullet$$ C)
• 2. A $$\supset$$ (F $$\bullet$$ E)
• 3. E / F

C.

• 1. (~F $$\vee$$ M) $$\supset$$ (P $$\vee$$ B)
• 2. (F $$\supset$$ P)
• 3. ~P / B

D.

• 1. M $$\supset$$ (F$$\bullet$$G)
• 2. (F $$\supset$$ K)
• 3. W
• 4. (W $$\supset$$ M) / K

E.

• 1. (M$$\supset$$F) $$\bullet$$ (Z$$\supset$$W)
• 2. (K $$\bullet$$ L) $$\bullet$$ A
• 3. K$$\supset$$ (M $$\vee$$ Z) / (F $$\vee$$ W)

F.

• 1. (M$$\supset$$F) $$\bullet$$ L
• 2. (F$$\supset$$G) $$\bullet$$ A
• 3. (M$$\supset$$G) $$\supset$$ [(M$$\supset$$F) $$\supset$$W] / W

G.

• 1. (M $$\bullet$$ F) $$\vee$$ (G $$\bullet$$ W)
• 2. (M $$\bullet$$ F) $$\supset$$L
• 3. (~L $$\bullet$$ A)
• 4. G $$\supset$$ (N $$\bullet$$ O) / N

H.

• 1. (F $$\bullet$$ A) $$\supset$$ (G $$\bullet$$ K)
• 2. (M$$\supset$$F) $$\bullet$$ G
• 3. (M $$\bullet$$ L)
• 4. (M$$\supset$$F) $$\supset$$ A / (G $$\vee$$ W)

I.

• 1. (M $$\supset$$ P)
• 2. (M $$\bullet$$ R)
• 3. P $$\supset$$ (Q $$\bullet$$ S)
• 4. (P$$\bullet$$Q) $$\supset$$ (P$$\equiv$$Q) / (P$$\equiv$$Q)

J.

• 1. A $$\supset$$ (B$$\supset$$C)
• 2. (A $$\bullet$$ M)
• 3. B $$\bullet$$ (F$$\bullet$$G) / (C $$\bullet$$ A)
##### Exercise $$\PageIndex{4}$$: Using all of the Rules of Implication and Replacement

Derive the conclusion (after the slash) from the numbered premises using any rule we’ve learned in this text.

A.

• 1. (X $$\supset$$ ~B)
• 2. (D $$\vee$$ X)
• 3. B /D

B.

• 1. [Q $$\vee$$ (A $$\vee$$ C)]
• 2. $$\neg$$ C / (A $$\vee$$ Q)

C.

• 1. ( $$\neg$$ M$$\supset$$B) $$\bullet$$ ( $$\neg$$ N$$\supset$$Q)
• 2. $$\neg$$ (M $$\bullet$$ N) / [(B $$\vee$$ Q) $$\vee$$ $$\neg$$ Z]

D.

• 1. (~X $$\vee$$ Y)
• 2. (~Y $$\vee$$ Z)
• 3. (X $$\bullet$$ W) /(X$$\bullet$$Y) $$\bullet$$ Z

E.

• 1. (~X $$\supset$$ T)
• 2. (W $$\bullet$$ ~T)
• 3. (X $$\vee$$ Y) $$\supset$$ Z / (W$$\bullet$$Z)

F.

• 1. (X $$\bullet$$ ~Z)
• 2. (Y $$\vee$$ X) $$\supset$$ ~W /~(Z $$\vee$$ W)

G.

• 1. ~(X $$\vee$$ ~X) / Y

H.

• 1. (G $$\supset$$ E)
• 2. (H $$\supset$$ ~E) / (G $$\supset$$ ~H)

I.

• 1. (~N $$\vee$$ B)
• 2. (N $$\supset$$ B) $$\supset$$ T / T

J.

• 1. ~X
• 2. ~(B $$\bullet$$ Q)
• 3. (~X $$\supset$$ B) / ~Q

K.

• 1. (Y $$\bullet$$ Z) $$\supset$$ X
• 2. (~X $$\bullet$$ ~Z)
• 3. Y $$\supset$$ (Y $$\bullet$$ Z) / (Z $$\vee$$ ~Y)

L.

• 1. (F $$\supset$$ ~J)
• 2. H $$\supset$$ (F $$\vee$$ G)
• 3. (G $$\supset$$ ~K)
• 4. H / ~(J$$\bullet$$K) $$\bullet$$ (H $$\vee$$ F)

M.

• 1. (A $$\supset$$ ~~B)
• 2. A $$\vee$$ (~C $$\vee$$ ~E)
• 3. ~B
• 4. D $$\supset$$ (C $$\bullet$$ E) / ~D

N.

• 1. (A $$\vee$$ M)
• 2. (A $$\supset$$ M) $$\bullet$$ (A $$\supset$$ O)
• 3. (M $$\supset$$ O)
• 4. (M $$\supset$$ B) $$\bullet$$ (O $$\supset$$ A) / (O $$\vee$$ B)

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