7.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.

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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)

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)

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.

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

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

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

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

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

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

1. ~(X \(\vee\) ~X) / Y

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

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

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

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

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

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

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)