2.1 Logical Consequence

A	B	$A \to B$	$A \land (A \to B)$
0	0	1	0
0	1	1	0
1	0	0	0
1	1	1	1
Statement 1 is proved

A	B	$A \to B$	$\neg A \to \neg B$
0	0	1	1
0	1	1	0
1	0	0	1
1	1	1	1
Statement 2 is disproved

A	B	$A \to B$	$(A \to B) \lor (B \to A)$
0	0	1	1
0	1	1	1
1	0	0	1
1	1	1	1
Statement 3 is proved

$A$	$B$	$C$	$A \to B$	$B \to C$	$(A \to B) \land (B \to C)$	$A \to C$
0	0	0	1	1	1	1
0	0	1	1	1	1	1
0	1	0	1	0	0	1
0	1	1	1	1	1	1
1	0	0	0	1	0	0
1	0	1	0	1	0	1
1	1	0	1	0	0	0
1	1	1	1	1	1	1
Statement 4 is proved

2.2 Satisfiability and Tautologies

satisfiable and not a tautology,
- example satisfiable: $A = 0$ $B = 1$
- example not a tautology: $A = 1$ $B = 1$
satisfiable and tautology: proved by the table in exercise 2.1.4

F = (B \to A) \land \neg ((\neg A \land \neg C) \land (\neg C \lor B))

$G = A \lor (\neg B \land \neg\neg C)$ Proof:

$F \equiv (\neg B \lor A) \land \neg ((\neg A \land \neg C) \land (\neg C \lor B))$ Definition of $\to$
$\equiv (\neg B \lor A) \land \neg (\neg A \land (\neg C \land (\neg C \lor B)))$ associativity of $\land$
$\equiv (\neg B \lor A) \land \neg (\neg A \land \neg C)$ absorption
$\equiv (\neg B \lor A) \land (\neg\neg A \lor \neg\neg C)$ de Morgan’s rule
$\equiv (\neg B \lor A) \land (A \lor \neg\neg C)$ double negation
$\equiv (A \lor \neg B) \land (A \lor \neg\neg C)$ commutativity of $\lor$
$\equiv A \lor (\neg B \land \neg\neg C)$ second distributive law $F \equiv G$ is proved

Does the left road lead to the village if and only if you are a knight?

A	$B$	$A \leftrightarrow B$	Answer
0	0	1	0
0	1	0	0
1	0	0	1
1	1	1	1

$F = A \leftrightarrow B$

$0 < n \cdot m ⊨ (0 < n) \lor (0 < m)$ false
$\forall n \exists m \exists k (n < m \land m = 3 \cdot k)$ true
$\forall x \exists n \exists m (p r im e (n) \land p r im e (m) \land (x = n + m))$ ???possibly true

For all integer x there is a integer reciprocal of x (false)
There exists an x such that x does not form a product of 1 with any y, and there exists a positive y (true)