1 CSCI-2200 FOUNDATIONS OF COMPUTER SCIENCE Spring 2019
2 Announcements • Homework 2 is due Thursday, Jan. 31 at 11:59pm. • Solutions to Homework 1 will be presented in recitation this week.
3 MATHEMATICAL PROOFS Sections 1.7 – 1.8
4 Recap - Proving Theorems • Many theorems have the form: • To prove them, we show that, where c is an arbitrary element of the domain, • By universal generalization the truth of the original formula follows. • So, we must prove something of the form: prove p → q • Direct Proof : To prove p → q is true: • Assume that p is true. • Use rules of inference, axioms, and logical equivalences to show that q must also be true. Proof by Contraposition : To prove p → q is true, • • Assume ¬q • Show ¬p is also true.
5 Proof Strategies for proving p → q • Choose a method. First try a direct method of proof. 1. If this does not work, try an indirect method (e.g., try to 2. prove the contrapositive). For whichever method you are trying, choose a strategy. • First try forward reasoning . Start with the axioms and 1. known theorems and construct a sequence of steps that end in the conclusion. Start with p and prove q, or start with ¬q and prove ¬p. If this doesn’t work, try backward reasoning . When 2. trying to prove q, find a statement p that we can prove with the property p → q .
6 Recap - Proof by Contradiction Proof by Contradiction : (AKA reductio ad absurdum ): • Suppose we want to prove that a proposition p is true. • First, assume that p is false (¬p is true). • Then, show that ¬p implies a contradiction, i.e., ¬p → (r ∧ ¬r) for some proposition r. • This means that ¬p → F is true. • It follows that the contrapositive T → p is true. • Therefore, p is true.
7 Recap - Definitions • A real number r is rational if r = p/q for some integers p and q, q ≠ 0, where p and q have no common factors other than 1. • It can also be written as r = k(p/q), where r is any positive integer. • A real number that is not rational is called irrational .
8 Recap – Predicate Logic • Theorem: The difference between any rational number and any irrational number is irrational.
9 • Theorem: The difference between any rational number and any irrational number is irrational.
10 • Theorem: The difference between any rational number and any irrational number is irrational. • Corollary: The sum of a rational number and an irrational number is irrational. • You can use this theorem and corollary in your homework. • What about this one? • Theorem: The product of a non-zero rational number and an irrational number is irrational. • You may need to prove this for HW2.
11 Give a proof by contradiction for “√2 is irrational”.
12 What is wrong with this? “Proof” that 1 = 2
13 Theorems that are Biconditional Statements • To prove a theorem that is a biconditional statement, i.e., a statement of the form p ↔ q , we show that p → q and q → p are both true. • Example Theorem: “If n is an integer, then n is odd if and only if n 2 is odd.” • Must prove both: If n is odd, then n 2 is odd. If n 2 is odd, then n is odd. • Can use different proof methods for each conditional statement.
14 Proof by Cases • To prove a conditional statement of the form: • Use the tautology • Each of the implications is a case . • You need to prove each of them. • Can use different proof method for each case (though try to avoid this if you can)
15 Example - Proof by Cases • Let a @ b = max{a, b} If a ≥ b, a @ b = a. Otherwise a @ b = b. • Show that for all real numbers a , b , c (a @b) @ c = a @ (b @ c) *This means the operation @ is associative. • Proof : Let a , b , and c be arbitrary real numbers. Then one of the following 6 cases must hold. a ≥ b ≥ c 1. a ≥ c ≥ b 2. b ≥ a ≥ c 3. b ≥ c ≥ a 4. c ≥ a ≥ b 5. c ≥ b ≥ a 6.
16 • Show that for all real numbers a , b , c (a@b)@c = a@(b@c).
17 Without Loss of Generality • Show that if x and y are integers and both xy and x + y are even, then both x and y are even. • We will use a proof by contraposition. • What do we assume? • What do we want to prove?
18 • Show that if x and y are integers and both xy and x + y are even, then both x and y are even.
19 • Show that if x and y are integers and both xy and x + y are even, then both x and y are even.
20 Without Loss of Generality • For the proof of “Show that if x and y are integers and both xy and x + y are even, then both x and y are even.” We only cover the case where x is odd because the case where y is odd is similar. The phrase without loss of generality (w.l.o.g.) indicates this.
21 A note about writing proofs…. • If you are writing a direct proof, you do not not need to write that the proof is a direct proof. • If you are using a different proof method, it is common to state which proof method you are using at the beginning of the proof, e.g., • “This is a proof by contradiction.” • “This is a proof by contraposition.” • “This is a proof by induction.”
22 A note about writing proofs… • For a proof of p → q by contraposition, we write something like… “This is a proof by contraposition. Assume ¬q is true. Steps from ¬q to ¬p go here This implies that ¬p is also true. Therefore, if p is true, then q is true.” For this class, this sentence is optional
23 Existence Proofs • Some theorems are of the form • E.g., “There is a positive integer that can be written as the sum of cubes of two positive integers in two different ways: • Constructive existence proof : • Find an explicit value of c , for which P(c) is true. • Then is true by Existential Generalization (EG). • Proof : 1729 is such a number since 1729 = 10 3 + 9 3 = 12 3 + 1
24 Nonconstructive Existence Proofs • In a nonconstructive existence proof : • We prove that a c exists that makes P(c) true, but we don’t actually find what that c is. • One option: we assume no c exists for which P(c) is true and derive a contradiction.
25 Theorem: “There exist irrational numbers x and y such that x y is rational.”
26 Good Problems to Review • Section 1.7: 9, 13, 15, 17b, 27 • Section 1.8: 5, 7, 11, 15
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