MATH1121 1-1

Introduction to Sets

  • Sets are collections of objects, essential for mathematics and later concepts like functions.

  • An element is any object within a set.

  • To denote membership, we use the symbol  (in set) and its crossed-out version for non-membership.

  • Order of elements in a set doesn’t matter; duplicates are not allowed.

Types of Sets

Natural Numbers (N)

  • Defines all whole positive numbers starting from 1 (1, 2, 3,...).

  • Notably, this definition excludes zero in this context.

  • Notation: N or double line when handwritten.

Integers (Z)

  • Comprises all whole numbers including zero and negative numbers (e.g., ...,-3,-2,-1,0,1,2,3,...).

  • The symbol Z derives from the German word "zahlen" (meaning numbers).

  • Notation: Z or double line when handwritten.

Rational Numbers (Q)

  • Any number that can be expressed as a fraction a/b where a and b are integers; b ≠ 0.

  • This set includes all fractions and whole numbers (e.g., 2 can be written as 2/1).

  • The symbol Q stands for "quotient" representing division.

  • Notation: Q or Q with a double line when handwritten.

Real Numbers (R)

  • Encompasses all rational numbers and irrational numbers (e.g., π, which cannot be expressed as a fraction).

  • Irrational numbers are those that can't be written as simple fractions.

  • Notation: R or real numbers written in a double line format.

Relationships Between Sets

  • Each set includes elements of the previous one, expanding the kinds of numbers considered.

  • Example: Two is an element of natural numbers (N), integers (Z), and real numbers (R).

  • π, although not a natural number or rational number, exists within the set of real numbers (R).

Intervals

  • An interval contains all real numbers between two endpoints (may include infinity).

  • Notation using square brackets [ ] includes endpoints and round brackets ( ) excludes them.

Examples of Intervals

  • For real numbers between 1 and 2, including endpoints: [1, 2]

  • For real numbers between 1 and 2, excluding endpoints: (1, 2)

  • Using mixed brackets: [1, 2) to include 1 but not 2.

  • Intervals can extend to infinity, e.g., [1, ∞) includes all real numbers from 1 onwards.

  • Remember: Infinity is not a number, thus uses round brackets.

Contexts of Usage

  • It’s important to distinguish between an interval and a coordinate point based on context.

  • For example, (1, 2) could denote either an interval or a point depending on context.

Example Evaluations

  1. Square Root of 2 in Integers:

    • False; √2 is not a whole number, hence not an integer.

  2. Number 6 in Real Numbers:

    • True; 6 is indeed a real number as it can be represented.

  3. Interval Notation for Numbers Greater than or Equal to 4:

    • False; correct notation should be [4, ∞) where ∞ cannot be included, so it should be (4, ∞) to avoid this mistake.