Proof and Elementary Number Theory A proof is

Proof and Elementary Number Theory • A proof is a logically rigorous and complete argument that a mathematical statement is true. • All mathematical proofs start with an implication or proposition and so it is necessary to establish whether the implication is true or false.

A statement is any sentence which is either true or false but not both e. g. a) b) c) 4 + 7 = 47 3 x + 4 x = 7 x 2+3>7 False True False

Negation The negation of the above statements are: a) b) c) 4 + 7 47 3 x + 4 x 7 x 2+3 7 True False True Notice how the True/False changes from the original statements. TJ 2 B Ex 1, page 85

SYMBOLS & NOTATION Universal Statements (‘for all’) 2 x + 1 is odd for all x Write: 2 x + 1 is odd x

Implications means ‘implies’ means that the first statement logically deduces the next statement. eg. For x є Z, if x > 3 then x 2 > 9 True Write x > 3 x 2 > 9

Implication: For x є Z, if x > 3 then x 2 > 9 (True) Related Implications: • Converse: If x 2 > 9 then x > 3 • Inverse: If x ≤ 3 then x 2 ≤ 9 • Contrapositive: If x 2 ≤ 9 then x ≤ 3 False True Note: If the original implication is true then the contrapositive is true, but the inverse and converse need not be true.

Further Examples Statement If P, then Q Converse If Q, then P Inverse If not P, then not Q Contrapositive If not Q, then not P Statement If triangle ABC is right angled at C, then c² = a² + b² Converse If c² = a² + b², then triangle ABC is right angled at C Inverse If triangle ABC is not right angled at C , then c² ≠ a² + b² Contrapositive If c² ≠ a² + b², then triangle ABC is not right angled at C

Statement If a quadrilateral is a rectangle, then it has two pairs TRUE of parallel sides. Converse If a quadrilateral has two pairs of parallel sides, then it FALSE is a rectangle. Inverse If a quadrilateral is not a rectangle, then it does not FALSE have two pairs of parallel sides. Contrapositive If a quadrilateral does not have two pairs of parallel TRUE sides, then it is not a rectangle. TJ 2 B Ex 2, page 86 – Converse Examples

Equivalence • means ‘equivalent to’ or ‘if and only if’ sometimes shortened to iff. It is a two-way implication e. g. a is even a 2 is even This would mean proving both a is even => a² is even AND a² is even => a is even • TJ 2 B Ex 3, page 86

Counterexample • An example that proves a statement is false. • Statement: x² + 1 is odd for all values of x • This is false since we can use the counterexample: • If x = 3, then 3² + 1 = 10, which is even. Book 2, page 3, Ex 1 A, Q 7 Further examples of all of above in Book 2, page 3, Ex 1 A

Direct Proof • Start with an accepted TRUE statement. • Use a logical set of steps • Arrive at a TRUE statement.

Examples – Direct Proof 1. If n is odd, then n² + 1 is even, n N 2. Prove that n² + 3 n is divisible by 2, n N • Worksheet • Book 2, page 5/6, Ex 1 B, Q 1 and 2 • Book 2, Ex 2 A, page 10/11, Q 1, 2, 3 a, b

Prove If n is odd, then n² + 1 is even, n N If n is odd, then n = 2 k +1, for some k N. Then, n² + 1 = (2 k+1)² + 1 = 4 k² + 4 k + 1 +1 = 4 k² + 4 k + 2 = 2(2 k² + 2 k + 1) = 2 x a number = even

Prove that n²+3 n is divisible by 2, n N • Assume n is even, n = 2 k, k N Then n²+3 n = (2 k)² + 3(2 k) = 4 k² + 6 k = 2(2 k² + 3 k) = 2 x an integer = even • Assume n is odd, n = 2 k + 1, k N, etc…. .

More on Direct Proof • Prove: If n² is odd, then n is odd • We get stuck. We can’t use a direct proof. • To prove P => Q, instead we prove the logically equivalent that (not Q) = > (not P), the contrapositive. • We assume NOT Q and prove NOT P

Proof by Contrapositive To prove P => Q we use a direct proof of not Q => not P. Examples Prove: If n² is odd, then n is odd • P=>Q • not Q => not P n² is odd => n is odd n is even => n² is even

Proof:

Proof by Contrapositive To complete the proof write: And so the contrapositive statement is true and it follows that the original statement “…”, that is equivalent to the contrapositive, is true. Class Exercise Prove: If n² is even, then n is even P=>Q not Q => not P n² is even => n is even n is odd => n² is odd

Proof:

Proof by Contrapositive Prove: If x and y are two integers for which x + y is even, then x and y are either both even or both odd. P=>Q: x + y even => x, y both odd or both even not Q => not P: x even and y odd => x + y odd (Should also show for x odd and y even)

Proof:

Proof by Contrapositive Exercise 1. If x and y are two integers whose product is even, then at least one of x and y must be even. 2. If x and y are two integers whose product is odd, then both must be odd. 3. Let x є Z. If 7 x + 9 is even, then x is odd. 4. Let x є Z. If x² - 6 x + 5 is even, then x is odd. 5. Let a, b, n є Z. If n does not divide ab, then n does not divide either a or b. If n ł ab, then n ł a and n ł b

Proof by Contradiction 1. If we are asked to prove P is true, make the assumption that P is not true and use implications to find a contradiction. 2. If we are asked to prove “if P then Q”, which is equivalent to “not Q then not P”, instead make the assumption that “if not Q then P” and continue to find a contradiction.

Proof by Contradiction: •

Proof:

Proof by Contradiction: Prove If n² is odd, then n is odd • Statement says: • n² odd => n odd P => Q • Negative Proposition: not Q => P • n even => n² odd

Proof:

Prove by Contradiction If x and y are integers and xy is an odd integer, prove that both x and y are odd. • Statement says: • xy odd => x and y both odd • Negative Proposition: • One of x and y even => xy odd Book 2, page 14, Ex 3 A, Qs 1, 2, 3, 6, 11, 12

Proof:

Proof by Induction • Often we have to prove for all values of n, which could take infinitely long. • Proof by Induction checks all values using just three steps.

Proof by Induction 1. Prove true for the first value, n = 1. 2. Assume true for n = k. Show that if it is true for n = k then it is also true for n = k + 1. 3. Conclusion - Hence, if true for n = k, then true for n = k + 1. But since true for n = 1, then by induction true for all positive integers n.

Prove, by Induction, 2 + 5 + 8 + …… + (3 n - 1) = ½n(3 n + 1)

Reminder – Summation Notation •

Prove, by Induction, •

• Photocopy Ex 5, p 85 from TJ 3 B Booklet

Further Examples on Induction • Book 2, Ex 4, page 20, Q 6, 7, 8, 9, 11, 12

The Division Algorithm & Number Bases

The following questions are similar to what appears in the unit assessment:

Euclidean Algorithm Use the Euclidean Algorithm to obtain the greatest common divisor of 1147 and 851

Counterexample For any real numbers a, b it is conjectured that if a + b is even then one of a or b is even • Use a counterexample to disprove this conjecture.