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\fQuantifiers . VxVy, P(x, y) = VyVx, P(x, y) . Fxly, P(x, y) = ]ylx, P(x, y) . Vxay, P(x, y) # Tyvx, P(x, y) - Vxly, P(x, y) # yVx, P(x, y) - 3yVx, P(x, y) = Vxly, P(x, y) . Vx(P(x.) AQ(2)) = (Vx, P(x)) A (Vx, Q(2)) . Vx (P(x) VQ(x)) # (Vx, P(x)) V (Vx, Q(x)) Let P(1) = Q(2) = True, P(2) = Q(1) = False, then LHS = True, RHS = False. . Fx (P(x) AQ(x)) # (3x, P(2)) A (3x, Q(2) ) Let P(1) = Q(2) = True, all other cases = False, then LHS = False, RHS = True. . x (P(2) VQ(T)) = (3x, P(x)) V (3x, Q(2)) Note 2 (Direct Proof, Proof by Contraposition, Proof by Contradiction, Proof by Cases) . Theorem 2.1: For any a, b, c E Z if a | b and a | c, then a | (b + c). . Theorem 2.2. Let 0 k, then at least one box must contain more than one object. . Theorem 2.6. There are infinitely many prime numbers. . Lemma 2.1. Every natural number greater than one is either prime or has a prime divisor. . Theorem 2.7. v2 is irrational. . Lemma 2.2. If a is even, then a is even. . Theorem 2.8. There exist irrational numbers x and y such that x3 is rational. Note 3 (Base Case, Inductive Hypothesis, Inductive Step) . Theorem 3.1: Vn E N, Zi = n(n+1) . Theorem 3.2: (Vn E N) (3 | (n3 -n)) . Theorem 3.3: Let P(n) denote the statement "Any map with n lines is two-colorable". Then, it holds that (Vn E N) (P(n))