Complete truth tables

Complete truth tables for each of the following:

1. $A\rightarrow A$

2. $C\rightarrow\neg C$

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

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

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

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

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

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

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

Check all the claims made in introducing the new notational conventions in §10.3, i.e. show that:

1. ‘$((A\wedge B)\wedge C)$’ and ‘$(A\wedge(B\wedge C))$’ have the same truth table

2. ‘$((A\vee B)\vee C)$’ and ‘$(A\vee(B\vee C))$’ have the same truth table

3. ‘$((A\vee B)\wedge C)$’ and ‘$(A\vee(B\wedge C))$’ do not have the same truth table

4. ‘$((A\rightarrow B)\rightarrow C)$’ and ‘$(A\rightarrow(B\rightarrow C))$’ do not have the same truth table

Also, check whether:

1. ‘$((A\leftrightarrow B)\leftrightarrow C)$’ and ‘$(A\leftrightarrow(B\leftrightarrow C))$’ have the same truth table

Write complete truth tables for the following sentences and mark the column that represents the possible truth values for the whole sentence.

1. $\neg(S\leftrightarrow(P\rightarrow S))$

2. $\neg[(X\wedge Y)\vee(X\vee Y)]$

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

4. $[C\leftrightarrow(D\vee E)]\wedge\neg C$

5. $\neg(G\wedge(B\wedge H))\leftrightarrow(G\vee(B\vee H))$

Write complete truth tables for the following sentences and mark the column that represents the possible truth values for the whole sentence.

1. $(D\wedge\neg D)\rightarrow G$

2. $(\neg P\vee\neg M)\leftrightarrow M$

3. $\neg\neg(\neg A\wedge\neg B)$

4. $[(D\wedge R)\rightarrow I]\rightarrow\neg(D\vee R)$

5. $\neg[(D\leftrightarrow O)\leftrightarrow A]\rightarrow(\neg D\wedge O)$

If you want additional practice, you can construct truth tables for any of the sentences and arguments in the exercises for the previous chapter.