Set, function, Polynomial function, Lecture notes of Mathematics

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IMAT Math

1. SET, FUNCTION, LINEAR FUNCTION DAVID. PARK

Finite Set : Has a limited number of elements, e.g. {1,2,3} Infinite Set : Continues without end, e.g., set of natural numbers {1,2,3,… } Empty Set (∅): Has no elements, e.g., set of square roots of negative numbers (in real numbers)

If an element 𝑥 belongs to a set A, we write: 𝑥 ∈ 𝐴 If an element 𝑥 does not belong to a set A, we write: 𝑥 ∉ 𝐴 If 𝐴 = 1 , 2 , 4 , then 2 ∈ 𝐴, but 3 ∉ 𝐴

Equal Sets : Have exactly the same elements, order does not matter (𝐴 = 𝐵) Universal Set (U): The set containing all elements under consideration

Union (A∪B): All elements in A or B or both Example: {1,2,3}∪{3,4}={1,2,3,4} Intersection (A∩B): Elements common to both A and B Example: {1,2,3}∩{3,4}={3} Difference (A−B): Elements in A but not in B Example: {1,2,3}−{3,4}={1,2} Complement (A′): Elements in the universal set but not in A

The overlapping region shows elements that belong to both sets (this represents the intersection , A∩B)

The entire area covered by the circles represents the union of sets (A∪B).

The number of elements in a set means how many things are in the set, we write the number of elements in set 𝐴 is 𝑛(𝐴) If 𝐴 = 1 , 2 , 4 , then 𝑛 𝐴 = 3 The empty set has no elements. i.e, 𝑛 ∅ = 0

When 𝐴 and 𝐵 be two sets, The Cartesian product of 𝐴 and 𝐵, denoted by 𝐴 × 𝐵 = 𝑎, 𝑏 𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵} The first component comes from set A, The second component comes from set B. Each element is an ordered pair.

1. Order matters: 𝑎, 𝑏 ≠ 𝑏, 𝑎 , 𝐴 × 𝐵 ≠ 𝐵 × 𝐴
2. The Cartesian product is not numerical multiplication. It is a way to construct a new set from two existing sets
3. Number of elements: 𝑛 𝐴 × 𝐵 = 𝑛(𝐴) × 𝑛(𝐵)

Let 𝐴 = 1 , 2 , 𝐵 = 𝑥, 𝑦, 𝑧 (1) List all elements of 𝐴 × 𝐵 (2) How many elements are in 𝐴 × 𝐵? (3) Is 𝐴 × 𝐵 = 𝐵 × 𝐴?

Fundamentals of functions Domain : All possible input values (x-values) Codomain : The set where output values come from (often ℝ) Range : Actual outputs (subset of codomain)

Fundamentals of functions A function is a rule that assigns exactly one output to each input. That is, x values does not repeat.