Your Turn
2.1
2.2
2.3
2.6
2.10
1.
If our friends did not come over to watch the game, then my roommates ordered pizza or I ordered wings.
2.11
3.
((\text{~}p) \vee (\text{~}q)) \leftrightarrow (\text{~}(p \wedge q)); this is another example of De Morgan’s Laws and it is always true.
2.12
2.15
2.16
3.
p | q | r | \text{~}q | p\, \wedge \text{~}q | (p \wedge \text{~}q) \vee r |
---|---|---|---|---|---|
T | T | T | F | F | T |
T | T | F | F | F | F |
T | F | T | T | T | T |
T | F | F | T | T | T |
F | T | T | F | F | T |
F | T | F | F | F | F |
F | F | T | T | F | T |
F | F | F | T | F | F |
2.17
2.18
2.19
1.
Valid
p | q | \text{~}{p} | \text{~}{p} \vee {q} | {q} \to \left( {\text{~}{p} \vee {q}} \right) |
---|---|---|---|---|
T | T | F | T | T |
T | F | F | F | T |
F | T | T | T | T |
F | F | T | T | T |
2.
Not valid
p | q | \text{~}{p} | {q} \wedge {p} | \text{~}{p} \to \left( {{q} \wedge {p}} \right) |
---|---|---|---|---|
T | T | F | T | T |
T | F | F | F | T |
F | T | T | F | F |
F | F | T | F | F |
2.20
2.21
1.
Valid
p | q | {p} \wedge {q} | \text{~}\left( {{p} \wedge {q}} \right) | \text{~}{p} | \text{~}{q} | \text{~}{p} \vee \text{~}{q} | \text{~}\left( {{p} \wedge {q}} \right) \leftrightarrow \left( {\text{~}{p} \vee \text{~}{q}} \right) |
---|---|---|---|---|---|---|---|
T | T | T | F | F | F | F | T |
T | F | F | T | F | T | T | T |
F | T | F | T | T | F | T | T |
F | F | F | T | T | T | T | T |
2.
Not Valid
p | q | \text{~}{p} | {q} \wedge {p} | \text{~}{p} \leftrightarrow \left( {{q} \wedge {p}} \right) |
---|---|---|---|---|
T | T | F | T | F |
T | F | F | F | T |
F | T | T | F | F |
F | F | T | F | F |
3.
Valid
p | q | {p} \to {q} | \text{~}{p} | \text{~}{p} \vee {q} | \left( {p \to q} \right) \leftrightarrow \left( {\text{~}p \vee q} \right) |
---|---|---|---|---|---|
T | T | T | F | T | T |
T | F | F | F | F | T |
F | T | T | T | T | T |
F | F | T | T | T | T |
4.
Valid
p | q | r | \text{~}{p} | \text{~}{q} | {p} \wedge {q} | \left( {{p} \wedge {q}} \right) \to {r} | \text{~}{p} \vee \text{~}{q} | \left( {{\text{~}p} \vee {\text{~}q}} \right) \vee {r} | \left( {{p} \wedge {q} \to {r}} \right) \leftrightarrow \left( {{\text{~}p} \vee {\text{~}q} \vee {r}} \right) |
---|---|---|---|---|---|---|---|---|---|
T | T | T | F | F | T | T | F | T | T |
T | T | F | F | F | T | F | F | F | T |
T | F | T | F | T | F | T | T | T | T |
T | F | F | F | T | F | T | T | T | T |
F | T | T | T | F | F | T | T | T | T |
F | T | F | T | F | F | T | T | T | T |
F | F | T | T | T | F | T | T | T | T |
F | F | F | T | T | F | T | T | T | T |
2.22
1.
p \to q is logically equivalent to \text{~}q \to\text{~}p.
p | {q} | {p} \to {q} | \text{~}{q} | \text{~}{p} | \text{~}{q} \to \text{~}{p} | \left( {p} \to {q} \right) \leftrightarrow \left( {\text{~}{q} \to \text{~}{p}} \right) |
---|---|---|---|---|---|---|
T | T | T | F | F | T | T |
T | F | F | T | F | F | T |
F | T | T | F | T | T | T |
F | F | T | T | T | T | T |
2.
p \to q is not logically equivalent to p \vee \text{~}q.
p | q | {p} \to {q} | \text{~}{q} | {p}\rm \vee \text{~}{q} | \left( {{p} \to {q}} \right) \leftrightarrow \left( {{p}{\rm{ }} \vee \text{~}{q}} \right) |
---|---|---|---|---|---|
T | T | T | F | T | T |
T | F | F | T | T | F |
F | T | T | F | F | F |
F | F | T | T | T | T |
2.23
2.25
2.26
2.27
2.28
2.29
1.
Eric needs to replace the light bulb, and Marcos did not leave the light bulb on all night, and Dan did not break the light bulb.
2.30
1.
p | q | {p} \vee {q} | \text{~}\left( {{p} \vee {q}} \right) | \text{~}p | \text{~}q | \text{~}p \wedge \text{~}q | \text{~}({p} \vee {q}) \leftrightarrow \left( {\text{~}p \wedge \text{~}q} \right) |
---|---|---|---|---|---|---|---|
T | T | T | F | F | F | F | T |
T | F | T | F | F | T | F | T |
F | T | T | F | T | F | F | T |
F | F | F | T | T | T | T | T |
2.31
2.32
2.33