Solutions to Chapter 11 Complete truth tables ‣ Solutions to selected exercices ‣ forall x: Calgary

A. Complete truth tables for each of the following:

1.

$A\rightarrow A$

$A$ $A$ $\rightarrow$ $A$

T T T T

F F T F
2.

$C\rightarrow\neg C$

$C$ $C$ $\rightarrow$ $\neg$ $C$

T T F F T

F F T T F
3.

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

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

T T T T T T T T F F T

T F T F F T F T T T F

F T F F T T F F T F T

F F F T F T T F F T F
4.

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

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

T T T T T T T T T

T F T F F T F T T

F T F T T T T F F

F F F T F T F T F
5.

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

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

T T T T T T T T T

T F T F F T F T T

F T F F T T T T F

F F F F F T F F F
6.

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

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

T T F T T T T F T F F T

T F F T T F T F T F T F

F T F F T T T T F F F T

F F T F F F T T F T T F

7.

$\bigl{[}(A\wedge B)\wedge\neg(A\wedge B)\bigr{]}\wedge C$

A

B

C

\bigl{[}(A

\wedge

B)

\wedge

\neg

(A

\wedge

B)\bigr{]}

\wedge

C

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

8.

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

$A$	$B$	$C$	$[(A$	$\wedge$	$B)$	$\wedge$	$C]$	$\rightarrow$	$B$
T	T	T	T	T	T	T	T	T	T
T	T	F	T	T	T	F	F	T	T
T	F	T	T	F	F	F	T	T	F
T	F	F	T	F	F	F	F	T	F
F	T	T	F	F	T	F	T	T	T
F	T	F	F	F	T	F	F	T	T
F	F	T	F	F	F	F	T	T	F
F	F	F	F	F	F	F	F	T	F

9.

$\neg\bigl{[}(C\vee A)\vee B\bigr{]}$

$A$ $B$ $C$ $\neg\bigl{[}($ $C$ $\vee$ $A)$ $\vee$ $B\bigr{]}$

T T T F T T T T T

T T F F F T T T T

T F T F T T T T F

T F F F F T T T F

F T T F T T F T T

F T F F F F F T T

F F T F T T F T F

F F F T F F F F F

B. 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

A

B

C

(A

\wedge

B)

\wedge

C

A

\wedge

(B

\wedge

C)

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

2.

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

A

B

C

(A

\vee

B)

\vee

C

A

\vee

(B

\vee

C)

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

3.

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

A

B

C

(A

\vee

B)

\wedge

C

A

\vee

(B

\wedge

C)

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

4.

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

A

B

C

(A

\rightarrow

B)

\rightarrow

C

A

\rightarrow

(B

\rightarrow

C)

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

Also, check whether:

5.

‘ $((A\leftrightarrow B)\leftrightarrow C)$ ’ and ‘ $(A\leftrightarrow(B\leftrightarrow C))$ ’ have the same truth table
Indeed they do:

A

B

C

(A

\leftrightarrow

B)

\leftrightarrow

C

A

\leftrightarrow

(B

\leftrightarrow

C)

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

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

1.

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

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

F T T T T T

F T T F T T

F F T T F F

T F F F T F
2.

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

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

F T T T T T T T

F T F F T T T F

F F F T T F T T

T F F F F F F F
3.

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

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

T T T T F T T F T

T F F T T F F F T

F T T F F T F T F

F T F T T F T T F
4.

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

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

T T T T T F F T

T T T T F F F T

T T F T T F F T

T F F F F F F T

F F T T T F T F

F F T T F F T F

F F F T T F T F

F T F F F T T F

5.

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

\neg

(G

\wedge

(B

\wedge

H))

\leftrightarrow

(G

\vee

(B

\vee

H))

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

T

F

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

1.

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

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

T F F T T T

T F F T T F

F F T F T T

F F T F T F
2.

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

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

F T F F T F T

F T T T F F F

T F T F T T T

T F T T F F F
3.

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

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

F T F T F F T

F T F T F T F

F T T F F F T

T F T F T T F

4.

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

[(D	$\wedge$	R)	$\rightarrow$	I]	$\rightarrow$	$\neg$	(D	$\vee$	R)
T	T	T	T	T	F	F	T	T	T
T	T	T	F	F	T	F	T	T	T
T	F	F	T	T	F	F	T	T	F
T	F	F	T	F	F	F	T	T	F
F	F	T	T	T	F	F	F	T	T
F	F	T	T	F	F	F	F	T	T
F	F	F	T	T	T	T	F	F	F
F	F	F	T	F	T	T	F	F	F

5.

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

$\neg$	[(D	$\leftrightarrow$	O)	$\leftrightarrow$	A]	$\rightarrow$	( $\neg$	D	$\wedge$	O)
F	T	T	T	T	T	T	F	T	F	T
T	T	T	T	F	F	F	F	T	F	T
T	T	F	F	F	T	F	F	T	F	F
F	T	F	F	T	F	T	F	T	F	F
T	F	F	T	F	T	T	T	F	T	T
F	F	F	T	T	F	T	T	F	T	T
F	F	T	F	T	T	T	T	F	F	F
T	F	T	F	F	F	F	T	F	F	F

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

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

$A$	$B$	$C$	$\neg\bigl{[}($	$C$	$\vee$	$A)$	$\vee$	$B\bigr{]}$
T	T	T	F	T	T	T	T	T
T	T	F	F	F	T	T	T	T
T	F	T	F	T	T	T	T	F
T	F	F	F	F	T	T	T	F
F	T	T	F	T	T	F	T	T
F	T	F	F	F	F	F	T	T
F	F	T	F	T	T	F	T	F
F	F	F	T	F	F	F	F	F