Solutions to Chapter 15 Partial truth tables

A. Use complete or partial truth tables (as appropriate) to determine whether these pairs of sentences are logically equivalent:

1. 1.
   ​

   $A$, $\neg A$ Not logically equivalent

   $A$	$A$	$\neg A$
   T	T	F

2. 2.
   ​

   $A$, $A\vee A$ Logically equivalent

   $A$	$A$	$A\vee A$
   T	T	T
   F	F	F

3. 3.
   ​

   $A\to A$, $A\leftrightarrow A$ Logically equivalent

   $A$	$A\to A$	$A\leftrightarrow A$
   T	T	T
   F	T	T

4. 4.
   ​

   $A\vee \neg B$, $A\to B$ Not logically equivalent

   $A$	$B$	$A\vee \neg B$	$A\to B$
   T	F	T	F

5. 5.
   ​

   $A\wedge \neg A$, $\neg B\leftrightarrow B$ Logically equivalent

   $A$	$B$	$A$	$\wedge$	$\neg$	$A$	$\neg$	$B$	$\leftrightarrow$	$B$
   T	T		F	F		F		F
   T	F		F	F		T		F
   F	T		F			F		F
   F	F		F			T		F

6. 6.
   ​

   $\neg (A\wedge B)$, $\neg A\vee \neg B$ Logically equivalent

   $A$	$B$	$\neg$	$(A$	$\wedge$	$B)$	$\neg$	$A$	$\vee$	$\neg$	$B$
   T	T	F		T		F		F	F
   T	F	T		F		F		T	T
   F	T	T		F		T		T	F
   F	F	T		F		T		T	T

7. 7.
   ​

   $\neg (A\to B)$, $\neg A\to \neg B$ Not logically equivalent

   $A$	$B$	$\neg$	$(A$	$\to$	$B)$	$\neg$	$A$	$\to$	$\neg$	$B$
   T	T	F		T		F		T	F

8. 8.
   ​

   $(A\to B)$, $(\neg B\to \neg A)$ Logically equivalent

   $A$	$B$	$(A\to B)$	$(\neg$	$B$	$\to$	$\neg$	$A)$
   T	T	T	F		T
   T	F	F	T		F	F
   F	T	T	F		T
   F	F	T	T		T	T

B. Use complete or partial truth tables (as appropriate) to determine whether these sentences are jointly satisfiable, or jointly unsatisfiable:

1. 1.
   ​

   $A\wedge B$, $C\to \neg B$, $C$ Jointly unsatisfiable

   $A$	$B$	$C$	$A\wedge B$	$C$	$\to$	$\neg B$	$C$
   T	T	T	T		F	F	T
   T	T	F	T		T		F
   T	F	T	F		T	T	T
   T	F	F	F		T		F
   F	T	T	F		F	F	T
   F	T	F	F		T		F
   F	F	T	F		T	T	T
   F	F	F	F		T		F

2. 2.
   ​

   $A\to B$, $B\to C$, $A$, $\neg C$ Jointly unsatisfiable

   $A$	$B$	$C$	$A\to B$	$B\to C$	$A$	$\neg C$
   T	T	T	T	T	T	F
   T	T	F	T	F	T	T
   T	F	T	F	T	T	F
   T	F	F	F	T	T	T
   F	T	T	T	T	F	F
   F	T	F	T	F	F	T
   F	F	T	T	T	F	F
   F	F	F	T	T	F	T

3. 3.
   ​

   $A\vee B$, $B\vee C$, $C\to \neg A$ Jointly satisfiable

   $A$	$B$	$C$	$A\vee B$	$B\vee C$	$C$	$\to$	$\neg A$
   F	T	T	T	T		T	T

4. 4.
   ​

   $A$, $B$, $C$, $\neg D$, $\neg E$, $F$ Jointly satisfiable

   $A$	$B$	$C$	$D$	$E$	$F$	$A$	$B$	$C$	$\neg D$	$\neg E$	$F$
   T	T	T	F	F	T	T	T	T	T	T	T

C. Use complete or partial truth tables (as appropriate) to determine whether each argument is valid or invalid:

1. 1.
   ​

   $A\vee [A\to (A\leftrightarrow A)]\therefore A$ Invalid

   $A$	$A$	$\vee$	$[A$	$\to$	$(A$	$\leftrightarrow$	$A)]$	$A$
   F		T		T				F

2. 2.
   ​

   $A\leftrightarrow \neg (B\leftrightarrow A)\therefore A$ Invalid

   $A$	$B$	$A$	$\leftrightarrow$	$\neg$	$(B\leftrightarrow A)$	$A$
   F	F		T	F	T	F

3. 3.
   ​

   $A\to B,B\therefore A$ Invalid

   $A$	$B$	$A\to B$	$B$	$A$
   F	T	T	T	F

4. 4.
   ​

   $A\vee B,B\vee C,\neg B\therefore A\wedge C$ Valid

   $A$	$B$	$C$	$A\vee B$	$B\vee C$	$\neg B$	$A\wedge C$
   T	T	T				T
   T	T	F			F	F
   T	F	T				T
   T	F	F	T	F	T	F
   F	T	T			F	F
   F	T	F			F	F
   F	F	T	F		T	F
   F	F	F	F		T	F

5. 5.
   ​

   $A\leftrightarrow B,B\leftrightarrow C\therefore A\leftrightarrow C$ Valid

   $A$	$B$	$C$	$A\leftrightarrow B$	$B\leftrightarrow C$	$A\leftrightarrow C$
   T	T	T			T
   T	T	F	T	F	F
   T	F	T			T
   T	F	F	F		F
   F	T	T	F		F
   F	T	F			T
   F	F	T	T	F	F
   F	F	F			T

D. Determine whether each sentence is a tautology, a contradiction, or a contingent sentence. Justify your answer with a complete or partial truth table as appropriate.

E. Determine whether each sentence is a tautology, a contradiction, or a contingent sentence. Justify your answer with a complete or partial truth table as appropriate.

1. 1.
   ​

   $\neg (A\vee A)$ Contingent

2. 2.
   ​

   $(A\to B)\vee (B\to A)$ Tautology

3. 3.
   ​

   $[(A\to B)\to A]\to A$ Tautology

4. 4.
   ​

   $\neg [(A\to B)\vee (B\to A)]$ Contradiction

5. 5.
   ​

   $(A\wedge B)\vee (A\vee B)$ Contingent

6. 6.
   ​

   $\neg (A\wedge B)\leftrightarrow A$ Contingent

7. 7.
   ​

   $A\to (B\vee C)$ Contingent

8. 8.
   ​

   $(A\wedge \neg A)\to (B\vee C)$ Tautology

9. 9.
   ​

   $(B\wedge D)\leftrightarrow [A\leftrightarrow (A\vee C)]$ Contingent

10. 10.
    ​

    $\neg [(A\to B)\vee (C\to D)]$ Contingent

F. Determine whether each the following pairs of sentences are logically equivalent using complete truth tables. Justify your answer with a complete or partial truth table as appropriate.

1. 1.
   ​

   $A$ and $A\vee A$

2. 2.
   ​

   $A$ and $A\wedge A$

3. 3.
   ​

   $A\vee \neg B$ and $A\to B$

4. 4.
   ​

   $(A\to B)$ and $(\neg B\to \neg A)$

5. 5.
   ​

   $\neg (A\wedge B)$ and $\neg A\vee \neg B$

6. 6.
   ​

   $((U\to (X\vee X))\vee U)$ and $\neg (X\wedge (X\wedge U))$

7. 7.
   ​

   $((C\wedge (N\leftrightarrow C))\leftrightarrow C)$ and $(\neg \neg \neg N\to C)$

8. 8.
   ​

   $[(A\vee B)\wedge C]$ and $[A\vee (B\wedge C)]$

9. 9.
   ​

   $((L\wedge C)\wedge I)$ and $L\vee C$

G. Determine whether each collection of sentences is jointly satisfiable or jointly unsatisfiable. Justify your answer with a complete or partial truth table as appropriate.

1. 1.
   ​

   $A\to A$, $\neg A\to \neg A$, $A\wedge A$, $A\vee A$ Consistent

2. 2.
   ​

   $A\to \neg A$, $\neg A\to A$ Insatisfiable

3. 3.
   ​

   $A\vee B$, $A\to C$, $B\to C$ Consistent

4. 4.
   ​

   $A\vee B$, $A\to C$, $B\to C$, $\neg C$ Insatisfiable

5. 5.
   ​

   $B\wedge (C\vee A)$, $A\to B$, $\neg (B\vee C)$ Insatisfiable

6. 6.
   ​

   $(A\leftrightarrow B)\to B$, $B\to \neg (A\leftrightarrow B)$, $A\vee B$ Consistent

7. 7.
   ​

   $A\leftrightarrow (B\vee C)$, $C\to \neg A$, $A\to \neg B$ Consistent

8. 8.
   ​

   $A\leftrightarrow B$, $\neg B\vee \neg A$, $A\to B$ Consistent

9. 9.
   ​

   $A\leftrightarrow B$, $A\to C$, $B\to D$, $\neg (C\vee D)$ Consistent

10. 10.
    ​

    $\neg (A\wedge \neg B)$, $B\to \neg A$, $\neg B$ Consistent

H. Determine whether each argument is valid or invalid. Justify your answer with a complete or partial truth table as appropriate.

1. 1.
   ​

   $A\to (A\wedge \neg A)\therefore \neg A$ Valid

2. 2.
   ​

   $A\vee B$, $A\to B$, $B\to A\therefore A\leftrightarrow B$ Valid

3. 3.
   ​

   $A\vee (B\to A)\therefore \neg A\to \neg B$ Valid

4. 4.
   ​

   $A\vee B$, $A\to B$, $B\to A\therefore A\wedge B$ Valid

5. 5.
   ​

   $(B\wedge A)\to C$, $(C\wedge A)\to B\therefore (C\wedge B)\to A$ Invalid

6. 6.
   ​

   $\neg (\neg A\vee \neg B)$, $A\to \neg C\therefore A\to (B\to C)$ Invalid

7. 7.
   ​

   $A\wedge (B\to C)$, $\neg C\wedge (\neg B\to \neg A)\therefore C\wedge \neg C$ Valid

8. 8.
   ​

   $A\wedge B$, $\neg A\to \neg C$, $B\to \neg D\therefore A\vee B$ Valid

9. 9.
   ​

   $A\to B\therefore (A\wedge B)\vee (\neg A\wedge \neg B)$ Invalid

10. 10.
    ​

    $\neg A\to B$,$\neg B\to C$,$\neg C\to A\therefore \neg A\to (\neg B\vee \neg C)$ Invalid

I. Determine whether each argument is valid or invalid. Justify your answer with a complete or partial truth table as appropriate.

1. 1.
   ​

   $A\leftrightarrow \neg (B\leftrightarrow A)\therefore A$ Invalid

2. 2.
   ​

   $A\vee B$, $B\vee C$, $\neg A\therefore B\wedge C$ Invalid

3. 3.
   ​

   $A\to C$, $E\to (D\vee B)$, $B\to \neg D\therefore (A\vee C)\vee (B\to (E\wedge D))$ Invalid

4. 4.
   ​

   $A\vee B$, $C\to A$, $C\to B\therefore A\to (B\to C)$ Invalid

5. 5.
   ​

   $A\to B$, $\neg B\vee A\therefore A\leftrightarrow B$ Valid