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8.E: Chapter Seven (Exercises)

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    223909
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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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