Exercise H

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