Mini-Introduction to Propositional Logic and Proofs

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1 Mini-Introduction to Propositional Logic and Proofs 1 Motivation: reasoning with flawed logic Solve for x: x + 2 = 4 x We get x = 5. Let s check the solution... The problem: we did not correctly identify statements and connectives (i) statements & connectives; (ii) proof strategies 2 Statements & Connectives, Quantifiers Statement: any assertion that is T or F sentence: Today, the sun shines. formula: for x it holds that p x w x < 1 non-example: 5x label: ϕ, ρ, σ ϕ : x < 1, or ρ : p x w c Ronald Wendner Logic & Proofs - 1 v2.1

2 statement typically expressed in terms of sets A, B A = {x R 1 < x < 1} B = {x R p x w}, for p, w R given ϕ same as x A ρ same as x B for a given x R, ϕ can by T or F; ρ can be T or F statements can be combined by 7 connectives/quantifiers,,,,,, let A, B X e.g., X = R and let p = w = 1 so that B = {x R x 1} and ϕ ρ or, equivalently, A B by A B we mean: (x A) (x B) to A B there is a corresponding set notation: A B statement A B is T if and only if A and B are both T c Ronald Wendner Logic & Proofs - 2 v2.1

3 truth values can be displayed by truth table A B A B T T T T F F F T F F F F ϕ ρ ϕ ρ T T T T F F F T F F F F equivalently, in set notation: Query. A B A B x A x B x (A B) x A x / B x / (A B) x / A x B x / (A B) x / A x / B x / (A B) (i) Show statements ϕ, ρ and sets A, B and (A B), as given above, graphically. (ii) Demonstrate all 4 cases given in the tables. c Ronald Wendner Logic & Proofs - 3 v2.1

4 or ϕ ρ or, equivalently, A B by A B we mean: (x A) (x B) to A B there is a corresponding set notation: A B statement A B is T if either A or B are both are T truth values A B A B T T T T F T F T T F F F ϕ ρ ϕ ρ T T T T F T F T T F F F equivalently, in set notation: A B A B x A x B x (A B) x A x / B x (A B) x / A x B x (A B) x / A x / B x / (A B) c Ronald Wendner Logic & Proofs - 4 v2.1

5 Query. (i) Show statements ϕ ρ and set (A B), as given above, graphically. (ii) Demonstrate all 4 cases given in the tables. not ϕ or, equivalently, A by A we mean: (x / A) to A there is a corresponding set notation: A c, where A c {x X x / A} is the set complement statement A is T if and only if A is F truth values A A ( A) T F T F T F A A c (A c ) c x A x / A c x (A c ) c x / A x A c x / (A c ) c Query. (i) Identify the sets A c and B c, as given above, analytically. (ii) Demonstrate the cases given in the table. c Ronald Wendner Logic & Proofs - 5 v2.1

6 implication ϕ ρ or, equivalently, A B A B means: for all x A, we have x B corresponding set notation: A B statement A B is T if either A is F or B is T x > 2 x 2 > 3 you participate in a doctoral program, therefore you are a student x is a square x is a rectangle if LHS is T then RHS must always be T but if RHS is T, LHS may or may not be T A sufficient not necessary condition for B but if B is F: then A must be F B required for A to be T B necessary not sufficient condition for A c Ronald Wendner Logic & Proofs - 6 v2.1

7 truth values A B A B T T T T F F F T T F F T Note: if A is F then A =, and is a subset of every set: (A = ) (A B), which is vacuously true for any B Query. (i) For the sets A and B, as given above, is A B true; is B A true? (ii) Show by means of a truth table that (A B) is the same as (A B). (iii) Show that the following statement is true: (y ) (y is a green-eyed lion) c Ronald Wendner Logic & Proofs - 7 v2.1

8 equivalence ϕ ρ or, equivalently, A B A B means: for all x A, we have x B and for all x B, we have x A corresponding set notation: A = B A is necessary and sufficient for B and B is necessary and sufficient for A: (A B) is the same as (A B) (B A) Query. Claim: (A B) [(A B) (B A)] Prove the claim by means of a truth table. A B A B A B B A (A B) (B A) T T T T T T T F F F T F F T F T F F F F T T T T Note. Truth values in column for A B follow from fact that A = B. Query. What went wrong with the motivating example? c Ronald Wendner Logic & Proofs - 8 v2.1

9 existential quantifier ( x A)(x B) set notation: A B F if A B = universal quantifier ( x A)(x B) set notation: A B F if ( x A)(x / B): counterexample! Query. Consider the sets A and B as defined at the beginning. (i) ( x A)(x B): T or F? (ii) ( x A)(x B): T or F? 3 Negation, again and, or, not (A B) ( A B) (A B) ( A B) ( A) A Query. Draw Venn diagrams to illustrate these cases. c Ronald Wendner Logic & Proofs - 9 v2.1

10 implication and equivalence (A B) (A B) (A B) [(A B) (B A)] (A B) [ (A B) (B A)] [(A B) (B A)] quantifiers ( x A)(x B) ( x A) (x B) ( x A)(x / B) (A B) = ( x A)(x B) ( x A) (x B) ( x A)(x / B) A B c Ronald Wendner Logic & Proofs - 10 v2.1

11 4 Proof strategies terminology we use tautology: (ρ ρ) always T contradiction: (ρ ρ) always F axiom theorem, proposition, corollary lemma definition, e.g., A c {x X x / A} proof strategies direct proof indirect proofs: by contrapositive, by contradiction, by induction steps involved 1. decide on a proof strategy 2. what is given information (LHS of ) 3. which statement exactly is to be shown (depending on proof strategy & given info) 4. try to find a way to show that statement, step by step (using given info) c Ronald Wendner Logic & Proofs - 11 v2.1

12 4.1 Direct Proof: (ϕ (ϕ ρ)) ρ objective: show A B strategy: (1) take A as given information (no need to show that A is true!) (2) show directly, step by step: A A 1 A 2... A n B Note. If we want to prove A B we need to prove [(A B) (B A)] Query. Prove the following statements by direct proofs. (i) ( x 2 + 5x 4 > 0) (x > 0) (ii) Let a, b, c, d R ++, and b > a. Then: (a c b d) (c > d) (iii) Let x R. (x 1) (2x 1)/x 1 What s the problem with (iii)? c Ronald Wendner Logic & Proofs - 12 v2.1

13 Cases I: A B We may split A into cases: A = A 1 A 2 (A B) [(A 1 A 2 ) B] [(A 1 B) (A 2 B)] Query. Let A = {x R 1 < x < 1} and B = {x R 0 < x < 1}. Employ a direct proof by cases to show: x A x 2 B. Cases II: A B We may split B into cases: (B 1 B 2 ) B [A (B 1 B 2 )] [(A B 1 ) (A B 2 )] Query. (i) Show by means of a truth table that the equivalence above is true. c Ronald Wendner Logic & Proofs - 13 v2.1

14 4.2 Proof by the Contrapositive objective: show A B strategy: (A B) ( B A). (1) take B as given information (no need to show that B is true!) (2) show that B... A Query. (i) Show, by means of a truth table, that the above equivalence is true. (ii) Employ a proof by the contrapositive to show ( x 2 + 5x 4 > 0) (x > 0). (iii) Let x R. Employ a proof by the contrapositive to show (x 1) [(2x 1)/x 1]. (iv) Let a, b be two integers. Employ a proof by the contrapositive to show [(ab) even] [(a even) (b even)]. c Ronald Wendner Logic & Proofs - 14 v2.1

15 4.3 Proof by Contradiction objective: show A B strategy: [(A B) (C C)] (A B) (1) suppose (A B), then (A B) (2) show (A B) (C C), a contradicition. (3) [ (C C) (A B)] (C C) is a tautology, and (A B) (A B) Query. By employing proofs by contradiction, show: (i) (a < b) (a + b)/2 < b; (ii) Suppose n is an integer. Then [(n 2 + 1) is odd] n is even. (iii) Let a, b be two integers. [(ab) even] [(a even) (b even)]. c Ronald Wendner Logic & Proofs - 15 v2.1

16 4.4 Proof by Induction objective: show ϕ(n) is T n N = {1, 2, 3,...} strategy: (1) base step: verify ϕ(1) that is, ϕ(n) is T for n = 1 (2) induction step: ϕ(k) ϕ(k + 1) Notice: in induction step, ϕ(k) is given info! [ϕ(1) ( k N)(ϕ(k) ϕ(k+1))] [( n N)(ϕ(n))] Query. Show that the following statements are true for all n N: (i) n i=1 i = n(1+n) 2 (ii) n i=0 2i = 2 n+1 1 (iii) n i=1 (2 i 1) = n2 c Ronald Wendner Logic & Proofs - 16 v2.1

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