Math 151 Unit 1 PDF

UNIT ONE MATH 151 UNIT ONE MATH 151 YAO ELIKEM AYEKPLE (PhD) KNUST January 17, 2024 1 / 31 UNIT ONE MATH 151 INTRO...

UNIT ONE MATH 151 UNIT ONE MATH 151 YAO ELIKEM AYEKPLE (PhD) KNUST January 17, 2024 1 / 31 UNIT ONE MATH 151 INTRODUCTION TO NUMBER THEORY 1.1 REAL NUMBERS (a) NATURAL NUMBER Let N denote the set of natural numbers. Thus N = {1, 2, 3,...}. The properties of natural numbers with addition (+) and multiplication (·) operations: A1 Closure Property of Addition For all a, b ∈ N, a + b ∈ N and if a = d and b = c, then a + b = d + c. A2 Commutative Property of Addition For all a, b ∈ N, a + b = b + a. 2 / 31 UNIT ONE MATH 151 A3 Associative property of addition For all a, b, c ∈ N, a + (b + c) = (a + b) + c. M1 Closure Property of Multiplication For all a, b ∈ N, a · b ∈ N and if a = d and b = c, then a · b = d · c. M2 Commutative Property of Multiplication For all a, b ∈ N, a · b = b · a. M3 Associative property of multiplication For all a, b, c ∈ N, a · (b · c) = (a · b) · c. M4 Multiplicative identity property There is a unique natural number 1 such that 1 ̸= 0 and a · 1 = a for all a ∈ N. 3 / 31 UNIT ONE MATH 151 DL Distributive property of multiplication over addition For all a, b, c ∈ N, a · (b + c) = a · b + a · c. 01 Trichotomy law For all a, b ∈ N, exactly one of the relations a = b, a > b, or a < b holds (trichotomy law). 02 Law of transitivity For all a, b, c ∈ N, if a < b and b < c, then a < c. 03 Closure under addition For all a, b, c ∈ N, if a < b, then a + c < b + c. 4 / 31 UNIT ONE MATH 151 (b) INTEGERS Let Z denote the set of integer numbers. Thus Z= {−3, −2, −1, 0, 1, 2,...}. The properties of integers with addition (+) and multiplication (·) operations: PROPERTIES OF INTEGERS A1 Closure Property of Addition For all a, b ∈ Z, a + b ∈ Z and if a = d and b = c, then a + b = d + c. 5 / 31 UNIT ONE MATH 151 A2 Commutative Property of Addition For all a, b ∈ Z, a + b = b + a. A3 Associative Property of Addition For all a, b, c ∈ Z, a + (b + c) = (a + b) + c. 6 / 31 UNIT ONE MATH 151 A4 Additive Identity Property There is a unique integer 0 such that a + 0 = a, for all a ∈ Z. A5 Additive Inverse Property For each a ∈ Z, there is a unique integer −a such that a + (−a) = 0. M1 Closure Property of Multiplication For all a, b ∈ Z, a · b ∈ Z and if a = d and b = c, then a · b = d · c. 7 / 31 UNIT ONE MATH 151 M2 Commutative Property of Multiplication For all a, b ∈ Z, a · b = b · a. M3 Associative Property of Multiplication For all a, b, c ∈ Z, a · (b · c) = (a · b) · c. M4 Multiplicative Identity Property There is a unique integer 1 such that 1 ̸= 0 and a · 1 = a for all a ∈ Z. DL Distributive Property of Multiplication over Addition For all a, b, c ∈ Z, a · (b + c) = a · b + a · c. 8 / 31 UNIT ONE MATH 151 01 Trichotomy Law For all a, b ∈ Z, exactly one of the relations a = b, a > b, or a < b holds (trichotomy law). 02 Law of transitivity For all a, b, c ∈ Z, if a < b and b < c, then a < c. 03 Closure Under Addition For all a, b, c ∈ Z, if a < b, then a + c < b + c. 04 Closure Under Multiplication For all a, b, c ∈ Z, if a < b and z > 0, then az < bz. 9 / 31 UNIT ONE MATH 151 (c) RATIONAL NUMBERS Let Q denote the set of rational numbers. Thus, n o p Q= q | p, q ∈ Z The properties of rational numbers with addition (+) and multiplication (·) operations: A1 Closure Property of Addition For all a, b ∈ Q, a + b ∈ Q and if a = d and b = c, then a + b = d + c. A2 Commutative Property of Addition For all a, b ∈ Q, a + b = b + a. 10 / 31 UNIT ONE MATH 151 A3 Associative Property of Addition For all a, b, c ∈ Q, a + (b + c) = (a + b) + c. A4 Additive Identity Property There is a unique rational number 0 such that a + 0 = a, for all a ∈ Q. 11 / 31 UNIT ONE MATH 151 A5 Additive Inverse Property For each a ∈ Q, there is a unique rational number −a such that a + (−a) = 0. M1 Closure Property of Multiplication For all a, b ∈ Q, a · b ∈ Q and if a = d and b = c, then a · b = d · c. M2 Commutative Property of Multiplication For all a, b ∈ Q, a · b = b · a. M3 Associative Property of Multiplication For all a, b, c ∈ Q, a · (b · c) = (a · b) · c. 12 / 31 UNIT ONE MATH 151 M4 Multiplicative Identity Property There is a unique rational number 1 such that 1 ̸= 0 and a · 1 = a for all a ∈ Q. M5 Multiplicative Inverse Property For each a ∈ Q with a ̸= 0, there is a unique rational number 1/a such that a · (1/a) = 1. 01 Trichotomy Law For all a, b ∈ Q, exactly one of the relations a = b, a > b, or a < b holds. 02 Law of Transitivity For all a, b, c ∈ Q, if a < b and b < c, then a < c. 13 / 31 UNIT ONE MATH 151 03 Closure Under Addition For all a, b, c ∈ Q, if a < b, then a + c < b + c. 04 Closure Under Multiplication For all a, b, c ∈ Q, if a < b and c > 0, then ac < bc. 14 / 31 UNIT ONE MATH 151 (D) IRRATIONAL NUMBERS Irrational numbers are numbers which are not rational, i.e., they cannot be expressed as ba ( called the quotient of a and b ) where a and b are √ integers and b ̸= 0.,eg. 2, π, e, etc. The set of rational and irrational numbers is called the set of real numbers,R. Well-Ordering Axiom Every nonempty subset of N contains a smallest element. 15 / 31 UNIT ONE MATH 151 1.2 The Principle of Mathematical Induction The principle of mathematical induction is an important property of the positive integers. Theorem (The Principle of Mathematical Induction) For each positive integer n, let P(n) be a statement. If 1 P(1) is true, and 2 the implication If P(k), then P(k + 1). is true for every positive integer k, then P(n) is true for every positive integer n. 16 / 31 UNIT ONE MATH 151 Alternatively, for each positive integer n, let P(n) be a statement. If 1 P(1) is true, and 2 ∀k ∈ N, P(k) =⇒ P(k + 1) is true, then ∀n ∈ N, P(n) is true. 17 / 31 UNIT ONE MATH 151 Examples Example 1 n(n+1) Prove that 1 + 2 + 3 +... + n = 2 for every positive integer n. Example 2 n(n+1)(2n+1) For every positive integer n, prove that 12 + 22 +... + n2 = 6. Example 3 For every positive integer n, prove that 1 1 1 n + +... + = 2·3 3·4 (n + 1)(n + 2) 2n + 4. 18 / 31 UNIT ONE MATH 151 A More General Principle of Mathematical Induction Theorem For a fixed integer m, let S = {i ∈ Z : i ≥ m}. For each integer n ∈ S, let P(n) be a statement. If 1 P(m) is true, and 2 the implication If P(k), then P(k + 1) is true for every integer k ∈ S, then P(n) is true for every integer n ∈ S. 19 / 31 UNIT ONE MATH 151 Alternatively, For a fixed integer m, let S = {i ∈ Z : i ≥ m}. For each n ∈ S, let P(n) be a statement. If 1 P(m) is true, and 2 ∀k ∈ S, P(k) =⇒ P(k + 1) is true, then ∀n ∈ S, P(n) is true. 20 / 31 UNIT ONE MATH 151 Questions Prove the following: Question 1 For every nonnegative integer n, 2n > n. Question 2 For every integer n ≥ 5, 2n > n 2. Question 3 For every nonnegative integer n, 3 | (22n − 1). 21 / 31 UNIT ONE MATH 151 Questions (cont’d) Question 4 For every nonnegative integer n, 9 | (43n − 1). Question 5 For every positive integer n, 6 | (n3 − n). Question 6 For every nonnegative integer n, 3 | (22n − 1). 22 / 31 UNIT ONE MATH 151 The Strong Principle of Mathematical Induction For each positive integer n, let P(n) be a statement. If 1 P(1) is true, and 2 the implication If P(i) for every integer i with 1 ≤ i ≤ k, then P(k + 1). is true for every positive integer k, then P(n) is true for every positive integer n. 23 / 31 UNIT ONE MATH 151 Alternatively, For each positive integer n, let P(n) be a statement. If 1 P(1) is true, and 2 ∀k ∈ N, P(1) ∧ P(2) ∧... ∧ P(k) =⇒ P(k + 1) is true, then ∀n ∈ N, P(n) is true. 24 / 31 UNIT ONE MATH 151 Questions Question 1 A sequence {an } is defined recursively by: a1 = 1, a2 = 3, and an = 2an−1 − an−2 for n ≥ 3. Then an = 2n − 1 for all n ∈ N. Question 2 A sequence {an } is defined recursively by: a1 = 1, a2 = 4, and an = 2an−1 − an−2 + 2 for n ≥ 3. Verify that your conjecture is correct. Question 3 A sequence {an } is defined recursively by: a1 = 1, a2 = 4, and an = 2an−1 − an−2 + 2 for n ≥ 3. Then an = n2 for all n ∈ N. 25 / 31 UNIT ONE MATH 151 Exercise 1 1.1 Which of the following sets are well-ordered? (a) S = {x ∈ Q : x ≥ −10} (b) S = {−2, −1, 0, 1, 2} (c) S = {x ∈ Q : −1 ≤ x ≤ 1} (d) S = {p : p is a prime} = {2, 3, 5, 7, 11, 13, 17,...} 1.2 Prove that 1 + 3 + 5 + · · · + (2n − 1) = n2 for every positive integer n, 1 by mathematical induction. 2 by adding 1 + 3 + 5 + · · · + (2n − 1) and (2n − 1) + (2n − 3) + · · · + 1. 26 / 31 UNIT ONE MATH 151 1.3 Use mathematical induction to prove that 1 + 5 + 9 + · · · + (4n − 3) = 2n2 − n for every positive integer n. 1.4 Find a formula for 1 + 4 + 7 + · · · + (3n − 2) for positive integers n, and then verify your formula by mathematical induction. 1.5 Find another formula suggested by Exercises 1.2 and 1.3, and verify your formula by mathematical induction. 27 / 31 UNIT ONE MATH 151 1.6 (a) We have seen that 12 + 22 +... + n2 is the number of squares in an n × n square composed of n2 1 × 1 squares. What does 13 + 23 + 33 +... + n3 represent geometrically? (b) Use mathematical induction to prove that 2 2 13 + 23 + 33 +... + n3 = n (n+1) 4 for every positive integer n. 1.7 Prove that n(n + 1)(2n + 7) 1 · 3 + 2 · 4 + 3 · 5 +... + n(n + 2) = 6 for every positive integer n. 28 / 31 UNIT ONE MATH 151 1.8 Let r ̸= 1 be a real number. Use induction to prove that a(1 − r n ) a + ar + ar 2 +... + ar n−1 = 1−r for every positive integer n. 1.9 Prove that 1 1 1 n + +... + = 3·4 4·5 (n + 2)(n + 3) 3n + 9 for every positive integer n. 1.10 Prove that 2n > n3 for every integer n ≥ 10. 1.11 Prove that n! > 2n for every integer n ≥ 4. 29 / 31 UNIT ONE MATH 151 1.12 Prove that 3n > n2 for every positive integer n. 1.13 Prove that 1 1 1 1 1+ + +... + 2 ≤ 2 − 4 9 n n for every positive integer n. 1.14 Prove Bernoulli’s Identity: For every real number x > −1 and every positive integer n, (1 + x)n ≥ 1 + nx. 1.15 Prove that 4 | (5n − 1) for every nonnegative integer n. 1.16 Prove that 81 | (10n+1 − 9n − 10) for every nonnegative integer n. 1.17 Prove that 7 | (32n − 2n ) for every nonnegative integer n. 30 / 31 UNIT ONE MATH 151 1.18 Use proof by minimum counterexample to prove that 6 | 7n(n2 − 1) for every positive integer n. 1.19 Use the method of minimum counterexample to prove that 3 | (22n − 1) for every positive integer n. 1.20 Prove that 12 | (n4 − n2 ) for every positive integer n. 1.21 Prove that 5 | (n5 − n) for every integer n. 1.22 Use proof by minimum counterexample to prove that 3 | (2n + 2n+1 ) for every nonnegative integer n. 30 / 31 UNIT ONE MATH 151 1.23 A sequence {an } is defined recursively by a1 = 1 and an = 2an−1 for n ≥ 2. Conjecture a formula for an and verify that your conjecture is correct. 1.24 A sequence {an } is defined recursively by a1 = 1, a2 = 2, and an = an−1 + 2an−2 for n ≥ 3. Conjecture a formula for an and verify that your conjecture is correct. 1.25 A sequence {an } is defined recursively by a1 = 1, a2 = 4, a3 = 9, and an = an−1 − an−2 + an−3 + 2(2n − 3) for n ≥ 4. Conjecture a formula for an and prove that your conjecture is correct. 31 / 31 UNIT ONE MATH 151 1.26 Consider the sequence F1 , F2 , F3 ,..., where F1 = 1, F2 = 1, F3 = 2, F4 = 3, F5 = 5, and F6 = 8. The terms of this sequence are called Fibonacci numbers. (a) Define the sequence of Fibonacci numbers by means of a recurrence relation. (b) Prove that 2 | Fn if and only if 3 | n. 31 / 31