Partial truth tables

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

1. $A$, $\neg A$

2. $A$, $A\vee A$

3. $A\rightarrow A$, $A\leftrightarrow A$

4. $A\vee\neg B$, $A\rightarrow B$

5. $A\wedge\neg A$, $\neg B\leftrightarrow B$

6. $\neg(A\wedge B)$, $\neg A\vee\neg B$

7. $\neg(A\rightarrow B)$, $\neg A\rightarrow\neg B$

8. $(A\rightarrow B)$, $(\neg B\rightarrow\neg A)$

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

1. $A\wedge B$, $C\rightarrow\neg B$, $C$

2. $A\rightarrow B$, $B\rightarrow C$, $A$, $\neg C$

3. $A\vee B$, $B\vee C$, $C\rightarrow\neg A$

4. $A$, $B$, $C$, $\neg D$, $\neg E$, $F$

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

1. $A\vee\bigl[A\rightarrow(A\leftrightarrow A)\bigr]\therefore A$

2. $A\leftrightarrow\neg(B\leftrightarrow A)\therefore A$

3. $A\rightarrow B,B\therefore A$

4. $A\vee B,B\vee C,\neg B\therefore A\wedge C$

5. $A\leftrightarrow B,B\leftrightarrow C\therefore A\leftrightarrow C$

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. $A\rightarrow\neg A$

2. $A\rightarrow(A\wedge(A\vee B))$

3. $(A\rightarrow B)\leftrightarrow(B\rightarrow A)$

4. $A\rightarrow\neg(A\wedge(A\vee B))$

5. $\neg B\rightarrow[(\neg A\wedge A)\vee B]$

6. $\neg(A\vee B)\leftrightarrow(\neg A\wedge\neg B)$

7. $[(A\wedge B)\wedge C]\rightarrow B$

8. $\neg\bigl[(C\vee A)\vee B\bigr]$

9. $\bigl[(A\wedge B)\wedge\neg(A\wedge B)\bigr]\wedge C$

10. $(A\wedge B)]\rightarrow[(A\wedge C)\vee(B\wedge D)]$

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.

1. $\neg(A\vee A)$

2. $(A\rightarrow B)\vee(B\rightarrow A)$

3. $[(A\rightarrow B)\rightarrow A]\rightarrow A$

4. $\neg[(A\rightarrow B)\vee(B\rightarrow A)]$

5. $(A\wedge B)\vee(A\vee B)$

6. $\neg(A\wedge B)\leftrightarrow A$

7. $A\rightarrow(B\vee C)$

8. $(A\wedge\neg A)\rightarrow(B\vee C)$

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

10. $\neg[(A\rightarrow B)\vee(C\rightarrow D)]$

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. $A\rightarrow A$, $\neg A\rightarrow\neg A$, $A\wedge A$, $A\vee A$

2. $A\rightarrow\neg A$, $\neg A\rightarrow A$

3. $A\vee B$, $A\rightarrow C$, $B\rightarrow C$

4. $A\vee B$, $A\rightarrow C$, $B\rightarrow C$, $\neg C$

5. $B\wedge(C\vee A)$, $A\rightarrow B$, $\neg(B\vee C)$

6. $(A\leftrightarrow B)\rightarrow B$, $B\rightarrow\neg(A\leftrightarrow B)$, $A\vee B$

7. $A\leftrightarrow(B\vee C)$, $C\rightarrow\neg A$, $A\rightarrow\neg B$

8. $A\leftrightarrow B$, $\neg B\vee\neg A$, $A\rightarrow B$

9. $A\leftrightarrow B$, $A\rightarrow C$, $B\rightarrow D$, $\neg(C\vee D)$

10. $\neg(A\wedge\neg B)$, $B\rightarrow\neg A$, $\neg B$

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

1. $A\rightarrow(A\wedge\neg A)\therefore\neg A$

2. $A\vee B$, $A\rightarrow B$, $B\rightarrow A\therefore A\leftrightarrow B$

3. $A\vee(B\rightarrow A)\therefore\neg A\rightarrow\neg B$

4. $A\vee B$, $A\rightarrow B$, $B\rightarrow A\therefore A\wedge B$

5. $(B\wedge A)\rightarrow C$, $(C\wedge A)\rightarrow B\therefore(C\wedge B)\rightarrow A$

6. $\neg(\neg A\vee\neg B)$, $A\rightarrow\neg C\therefore A\rightarrow(B\rightarrow C)$

7. $A\wedge(B\rightarrow C)$, $\neg C\wedge(\neg B\rightarrow\neg A)\therefore C\wedge\neg C$

8. $A\wedge B$, $\neg A\rightarrow\neg C$, $B\rightarrow\neg D\therefore A\vee B$

9. $A\rightarrow B\therefore(A\wedge B)\vee(\neg A\wedge\neg B)$

10. $\neg A\rightarrow B$,$\neg B\rightarrow C$,$\neg C\rightarrow A\therefore\neg A\rightarrow(\neg B\vee\neg C)$

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

1. $A\leftrightarrow\neg(B\leftrightarrow A)\therefore A$

2. $A\vee B$, $B\vee C$, $\neg A\therefore B\wedge C$

3. $A\rightarrow C$, $E\rightarrow(D\vee B)$, $B\rightarrow\neg D\therefore(A\vee C)\vee(B\rightarrow(E\wedge D))$

4. $A\vee B$, $C\rightarrow A$, $C\rightarrow B\therefore A\rightarrow(B\rightarrow C)$

5. $A\rightarrow B$, $\neg B\vee A\therefore A\leftrightarrow B$