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

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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)