# Mathematical Proof/Appendix/Answer Key/Mathematical Proof/Introduction/Logical Reasoning

```1.1, Logical Reasoning.
```

These are the answers to the exercises in section 1.1, Logical Reasoning

1. Truth tables
1. ${\displaystyle \lnot P\Rightarrow Q}$
P Q ${\displaystyle \lnot P}$ ${\displaystyle \lnot P\Rightarrow Q}$
T T F T
T F F T
F T T T
F F T F

2. ${\displaystyle P\Rightarrow \lnot Q}$
P Q ${\displaystyle \lnot Q}$ ${\displaystyle P\Rightarrow \lnot Q}$
T T F F
T F T T
F T F T
F F T T

3. ${\displaystyle (P\lor Q)\Rightarrow R}$
P Q R ${\displaystyle P\lor Q}$ ${\displaystyle (P\lor Q)\Rightarrow R}$
T T T T T
T T F T F
T F T T T
T F F T F
F T T T T
F T F T F
F F T F T
F F F F T

4. ${\displaystyle (P\land Q)\Rightarrow (R\lor S)}$
P Q R S ${\displaystyle P\land Q}$ ${\displaystyle R\lor S}$ ${\displaystyle (P\land Q)\Rightarrow (R\lor S)}$
T T T T T T T
T T T F T T T
T T F T T T T
T T F F T F F
T F T T F T T
T F T F F T T
T F F T F T T
T F F F F F T
F T T T F T T
F T T F F T T
F T F T F T T
F T F F F F T
F F T T F T T
F F T F F T T
F F F T F T T
F F F F F F T

5. ${\displaystyle (P\Rightarrow Q)\Leftarrow (R\Rightarrow S)}$
P Q R S ${\displaystyle P\Rightarrow Q}$ ${\displaystyle R\Rightarrow S}$ ${\displaystyle (P\Rightarrow Q)\Leftarrow (R\Rightarrow S)}$
T T T T T T T
T T T F T F T
T T F T T T T
T T F F T T T
T F T T F T F
T F T F F F T
T F F T F T F
T F F F F T F
F T T T T T T
F T T F T F T
F T F T T T T
F T F F T T T
F F T T T T T
F F T F T F T
F F F T T T T
F F F F T T T

2. Negated statements

1. ${\displaystyle \lnot P\lor \lnot Q}$
2. ${\displaystyle (\lnot P\land \lnot Q)\lor (\lnot R\lor \lnot S)}$
3. ${\displaystyle (\lnot P\lor Q)\land (R\land \lnot S)}$