Discrete-math 2

Prove: \neg (\forall a \exists b (a ∣ b \to ((a \leq b) \land (a + b < a * b))))

\begin{matrix} \exists a \forall b \neg ((a ∣ b \to ((a \leq b) \land (a + b < a * b)))) \\ \exists a \forall b (a ∣ b \land \neg ((a \leq b) \land (a + b < a * b))) \\ \exists a \forall b (a ∣ b \land (a > b \lor (a + b \geq a * b))) \\ \exists a \forall b (a ∣ b \land (a + b \geq a * b)) \\ \exists a = 1 : \forall b (T r u e \land (b + 1 \geq b)) \equiv T r u e \end{matrix}

\begin{matrix} P, Q over Z \\ Prove or disprove: \forall a ((P (a) \lor Q (a)) \to ((\forall a : P (a))) \lor (\forall a : Q (a))) \\ Let P (a) \equiv 2 ∣ a; Q (a) \equiv 2 ∤ a \\ \forall a : P (a) \equiv F a l s e; \forall a : Q (a) \equiv F a l s e \\ \forall a : P (a) \lor Q (a) \equiv T r u e \end{matrix}

\begin{matrix} Prove or disprove: (\forall a : P (a)) \lor (\forall a : Q (a)) \to \forall a : P (a) \lor Q (a) \\ Let (\forall a : P (a)) \lor (\forall a : Q (a)) \equiv T r u e \\ Let a \in Z \\ \forall a : P (a) ⟹ \forall a : P (a) \lor Q (a) \\ \forall a : Q (a) ⟹ \forall a : P (a) \lor Q (a) \\ ⟹ (\forall a : P (a)) \lor (\forall a : Q (a)) \to \forall a : P (a) \lor Q (a) \end{matrix}

Prove: {6 n ∣ n \in N} \subseteq {2 n ∣ n \in N}

\begin{matrix} \forall x \in B : x = 6 m = 2 * 3 m \\ \exists n = 3 m : x = 2 n \\ n \in N ⟹ x \in A ⟹ A \subseteq B \end{matrix}

\begin{matrix} A = {X | X \in N} \\ Prove: B = {1, 2, 7} \in A \end{matrix}

\begin{matrix} \forall x \in B : x \in N ⟹ B \subseteq N ⟹ B \in A \end{matrix}