# 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. $\lnot P \Rightarrow Q$
P Q $\lnot P$ $\lnot P \Rightarrow Q$
T T F T
T F F T
F T T T
F F T F

2. $P \Rightarrow \lnot Q$
P Q $\lnot Q$ $P \Rightarrow \lnot Q$
T T F F
T F T T
F T F T
F F T T

3. $(P\lor Q)\Rightarrow R$
P Q R $P\lor Q$ $(P\lor Q)\Rightarrow R$
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. $(P\land Q)\Rightarrow (R\lor S)$
P Q R S $P\land Q$ $R\lor S$ $(P\land Q)\Rightarrow (R\lor S)$
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. $(P\Rightarrow Q)\Leftarrow(R\Rightarrow S)$
P Q R S $P\Rightarrow Q$ $R\Rightarrow S$ $(P\Rightarrow Q)\Leftarrow (R\Rightarrow S)$
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. $\lnot P \lor \lnot Q$
2. $(\lnot P \land \lnot Q) \lor (\lnot R\lor \lnot S)$
3. $(\lnot P \lor Q)\land (R\land \lnot S)$