Mathematical Statements and Proofs Lecture 1 (Handout)

MOE Advanced Level Higher 3 (H3) Math Mathematical Statements and Proofs Lecture 1 1 / 54

Telegram Chat https://t.me/+My2fLlPuAUxlYjY1 2 / 54

Content 1 Types of (Mathematical) statements. 2 Direct Proof 3 Disprove: Counter-examples 4 Considering Case 5 Number Theory 3 / 54

Some Nomenclature and Notations The set of real numbers is denoted as R. Unless stated otherwise, the universal set of numbers will be the set of real numbers. Z = {0, ±1, ±2, ±3, ±4, . . .}: the set of integers. The set of numbers that can be obtained from 0 by adding or subtracting 1. Z+ = Z>0 = {1, 2, 3, 4, 5, . . .}: the set of positive integers. N = Z+, the set of natural numbers. ( Remark. Some textbooks includes 0 in N.) Q = {m/n | m, n∈ Z, n̸= 0}: the set of rational numbers. ∅: the empty set, the set containing no element. Similarly we can use Z−, Q+, Q−, R+, R−, . . . . C denoted the set of all complex numbers. Not discussed (formally). 4 / 54

Types of Statements (Structure) Conditional statements (i) In standard form: If P then Q (ii) Other forms: Q is necessary for P , or P is sufficient for Q (iii) Symbolic form: P ⇒ Q Here P is called the hypothesis or assumption; Q is called the conclusion . No assumption that P is true. But when P is true, then Q is. Example If n >2, then n >1. 5 / 54

Type of Statements (Structure) Biconditional Statements (i) Standard form: P if and only if Q (ii) Abbreviation: P iff Q (iii) Other forms: P is necessary and sufficient for Q (iv) Symbolic form: P ⇔ Q (v) Means P ⇒ Q and Q ⇒ P . Example n is odd if and only if n + 1is even. 6 / 54

Types of Statements (Structure) Existential Statement (i) Standard form: There exists an x in D such that P (x) (ii) Abbreviation: ∃ x ∈ D, P(x) (iii) D is the domain (iv) Other forms: there is a ..., there exists ..., for some ..., we can find ..., ... has ... Example There exists a real number x such that x2 = 2. There is a tallest person in the world. 7 / 54

Type of Statements (Content) Definition A (mathematical) definition is a (true) mathematical statement that gives the precise meaning of a word or phrase that represents some object, property or other concepts. To state a mathematical statement, we first need definitions. “All zaxcillian has rizz” makes no sense. Example (Negative Example) From https://www.dictionary.com/. Definition of happy: being delighted, pleased, or glad, as over a particular thing. Definition of pleased: very happy or satisfied Examples: It’s like when you win a match, or did well in your exam, or have an ice cream, ... etc. 8 / 54

Type of Statements (Content) Definition A (mathematical) definition is a (true) mathematical statement that gives the precise meaning of a word or phrase that represents some object, property or other concepts. Example An integer a is even if there exists an integer n such that a = 2n. An integer a is odd if there exists an integer n such that a = 2n + 1. 9 / 54

Type of Statements (Content) Theorem A theorem is a true mathematical statement that can be proven mathematically. Example n is odd if and only if n + 1is even. Axiom An axiom is a mathematical statement that does not require proof. Example If x and y are integers, then so is x + y. 10 / 54