MA1200: Sets and Functions

• Lecture note 2 covers the topic of sets and functions in the context of MA1200 Calculus and Basic Linear Algebra.

Set Notation

• A set A is a collection of distinct objects, which can include numbers, letters, or other entities.
• An object within a set is referred to as an element of the set A.
• A = {1, 3, 5, 7, 9} represents the set of all odd numbers between 1 and 10.
• B = {1, 2, 3, 4, 5, ...} represents the set of all positive integers.
• C = {0, +3, -3, +6, -6, +9, -9, ...} represents the set of multiples of 3.
• D = {all real numbers} = ℝ represents the set of real numbers.
• p ∈ B signifies that the element p is in the set B. The symbol "∈" means "belongs to".
• p ∉ B signifies that the element p is NOT in set B.
• For example: if E = {2, 3, 4, 5}, then 3 ∈ E and √6 ∉ E.
• Two sets A and B are equal only when both sets contain the same elements, written as A = B.
• If A = {1, 2, 3} and B = {1, 2, 3}, then A = B.
• If A = {1, 3, 4} and B = {1, 2, 3}, then A ≠ B.
• Given two sets A and B, A is a subset of B (denoted as A ⊆ B) if every element in A is also an element in B.
• If A = {1, 3} and B = {0, 1, 3, 4}, then A ⊆ B.
• If A = {2, 4} and B = {0, 1, 3, 4}, then A ∉ B.
• A general description of sets involves mentioning the common properties of the objects in the set, represented as E = {x | x has certain properties}.
• A = {x | x is prime and 0 < x ≤ 10} = {2, 3, 5, 7}.
• B = {x | x > 0 and x is a multiple of 3} = {3, 6, 9, 12, ...}.
• C = {x | x² ≤ 100 and x is a negative integer} = {x | -10 ≤ x ≤ 10 and x is a negative integer} = {-10, -9, -8, -7, -6, -5, -4, -3, -2, -1}.
• Ø represents the empty set or a set containing nothing.
• ℕ = {1, 2, 3, 4, 5, ...} represents the set of all positive integers.
• ℤ = {..., -3, -2, -1, 0, 1, 2, 3, ...} is defined as the set of all integers.
• ℚ = {m/n | m ∈ ℤ, n ∈ ℕ} is known as the set of rational numbers.
• [a, b] = {x | a ≤ x ≤ b}, [a, b) = {x | a ≤ x < b}, and (a, ∞) = {x | x > a} denote intervals.
• ℝ is the set of real numbers.
• ℂ represents the set of all complex numbers.
• "x ∈ ℝ" signifies that “x is real”, “x ∈ ℕ” signifies that “x is a positive integer,” and “x ∈ [a, b]" signifies that “a ≤ x ≤ b” or “x lies between a and b”.

Operations on sets

• The union of two sets, denoted by A ∪ B, is defined as A ∪ B = {x | x ∈ A or x ∈ B}.
• The intersection of two sets, denoted by A ∩ B, is defined as A ∩ B = {x | x ∈ A and x ∈ B}.
• The complement of B in A, denoted by A\B, is defined as A\B = {x | x ∈ A and x ∉ B}.
• [2, 8] ∪ [3, 10) = [2, 10).
• (3, 7) ∩ ℕ = {x | 3 < x < 7 and x is a positive integer} = {4, 5, 6}.
• Given that ℕ = {1, 2, 3, 4, ... } and ℤ = {..., -2, -1, 0, 1, 2, 3, 4, ... }, since every element in ℕ is also in ℤ, then ℕ\ℤ = Ø.

Functions

• A function f(x) from set A to set B is assigned (maps) each element of A to exactly one element of B.
• This is denoted by f: A → B where f(x) = y, with x ∈ A and y ∈ B.
• The domain of a function is the collection of numbers that can be “put” into the function.
• The codomain of a function is the set in which all possible outputs of the function f(x) lie.
• The range of a function is the collection of all possible outputs of the function.
• The range of f(x) does not necessarily cover the whole codomain.
• Examples of functions include:
  • f₁: ℝ → ℝ, given by f₁(x) = 2x
  • f2: {1, 2, 3, ... } → [1, ∞), given by f2(x) = 3x
  • f3: [0, ∞) → [0, ∞), given by f3(x) = √x. Here, √x takes zero or positive values.
• Examples of non-functions include:
  • 1
  • g₁(x) = (x-1)(x-3)
  • Its not a function because g₁(1) = 1/0 and g₁(3) = 1/0 which are not defined.
  • g2: ℝ → (-∞, 2), given by g2(x) = 4 – x².
  • Its not a function because g2(1) = 4 − (1)² = 3 which does not lie in the codomain (-∞, 2).

Range of a Function - Examples

• For g: ℕ → ℝ given by g(x) = 3x, the range of g(x) is {3, 6, 9, 12, 15, ... }, which is the positive multiple of 3.
• For h: ℝ → ℝ, given by h(x) = sin x, the range of h(x) is the interval [-1, 1].

Finding largest possible domain of the following functions

• f₁ (x) = x² — 2x – 3 can be calculated for every real number x, thus the domain of f₁ = ℝ. 1
• f2(x) = x²-2x-3 is not defined when x² – 2x − 3 = 0. -x² - 2x - 3 = 0 ⇒ (x - 3)(x + 1) = 0 ⇒ x = 3 or x = -1.
  • The domain of f₂ = ℝ{-1, 3}. x²-1
• f3 (x) = x-1 is not defined when x – 1 = 0, i.e. x = 1. -The domain of f₃ = ℝ{1}.
• f₄(x) = √4 – x² is defined only when 4 – x² ≥ 0, i.e. −2 ≤ x ≤ 2.
  • The domain of f₄ = [-2, 2].

Basic Operations of a function

• (f ± g)(x) = f(x) ± g(x) (addition and subtraction)
• (fg)(x) = f(x) × g(x) (multiplication)
• (f/g)(x) = f(x)/g(x) (division)
• (f ∘ g)(x) = f(g(x)) (composition). Note: (f ∘ g)(x) ≠ (g ∘ f)(x) in general.
• Let f(x) = x² + 1 and g(x) = 1 − x − x².
  • (f + g)(x) = f(x) + g(x) = x² + 1 + 1 − x − x² = 2 − x.
  • (fg)(x) = f(x)g(x) = (x² + 1)(1 − x − x²) = … = 1 − x − x³ – x⁴.
• Let f(x) = x² + 1 and g(x) = √x.
  • (f ∘ g)(8) = f(g(8)) = f(√8) = f(2) = 2² + 1 = 5.
  • (g ∘ f)(8) = g(f(8)) = g(8² + 1) = g(65) = √65 ≈ 4.02.
• Let f(x) = 100x and g(x) = log10 x
  • (f ∘ g)(x) = f(g(x)) = f(log x) = 100log10 x = 102log10 x = (10log10 x)² = x².
  • (g ∘ f)(x) = g(f(x)) = g(100x) = log10 100x = log10 102x = 2x log10 10 = 2x.

Commonly Used Functions

• Identity Function: I(x) = x. This function maps x to x itself.
• Constant Function: f(x) = c, where c is a fixed real number.
• The absolute value function, denoted by |x|.
  • |x| = x if x ≥ 0, -x if x < 0.
  • |5| = 5; |0| = 0; |-4| = -(-4) = 4.
  • Where |x|² = x²; |xy| = |x||y|; |x/y| = |x| / |y|
  • But |x + y| ≠ |x| + |y| and |x − y| ≠ |x| – |y| in general!
• Polynomial
  • A polynomial p(x) is a function of the following form: p(x) = anxn + an−1xn−1 + … + a₂x² + a₁x + a₀
  • n is a non-negative integer, and an, an−1,..., a0 are fixed numbers.
  • x² − 3x + 1 is a polynomial. x√2 − 2x and x−2 are NOT polynomials. x2+1 x3-x-1
• Rational: is rational but x+cos x/1-x5 is NOT rational.
• Six basic are trigonometric functions listed as: sin x, cos x, tan x, csc x= 1/sinx, sec x= 1/cosx, cot x= 1/tanx
• Exponential: f(x) = ax Where where a > 0 is constant and a ≠ 1 (When a = 1 X " f(x) = 1x = 1 which is a constant function).
• Logarithmic Function: y=log(a)x(x>0). Where a>0 is constant and a≠ 1 (base)
• Greatest Interger: denoted as [x], meaning greatest integer less than or equal to x. Example, [7.2]=7, [7.9] =7. Is discontinous or has a "jump" at integer points.

Monotone Functions

• Monotonic increasing function has f(x₁) ≤ f(x₂)
• Monotonic decreasing function has f(x₁) ≥ f(x₂)
• Strictly increasing function has f(x₁) < f(x₂)
• Strictly decreasing function has f(x₁) > f(x₂)

Periodic Functions

• Periodic Function is periodic with period T > 0 if f(x + T) = f(x).
• T should be the smallest number such that f(x + T) = f(x).
• Functions f(x) = sin x, g(x) = cos x are periodic with period 2π (or 360°).
• The function h(x) = sin 4x is periodic with period π/2 (or 90°).
• The function j(x) = 3x + 1 is not periodic.

Even and Odd Functions

• Even function is when f(-x) = f(x) for all x.
• Odd function is when f(-x) = -f(x) for all x.
• Even function graph is symmetric about the y-axis
• Odd function graph is symmetric about the origin
• The function f(x) = cos x is an even function since f(-x) = cos(-x) = cos x = f(x).
• The function f(x) = sin x is an odd function since f(-x) = sin(-x) = - sin x = -f(x).
• The function f(x) = (ax + a-x)/2 is an even function since f(-x) = (a-x + a-(-x))/2 = (a-x + ax)/2 = f(x).
• The function f(x) = x² + x − 1 is neither even nor odd.

Inverse Function

• Inverse of f^{-1}(f(x)) = x, which tries to retrieve x from f(x)
• The domain of f-1(x) is the range of f(x).
• Functions must be one-to-one, such that for every x_{1} in A there should be different element in B and exists if inverse f ⁻¹ exists. y+3
• To find inverse y=f(x) express in the terms of y.
• Example:
  y =f(x) = 2x -3 --> express x in terms of y --> x = (y+3)/2. Therefore final function is f^{-1} = 2

Transformations of Functions

• Graphically one can visualize function by observing a maximum, minimum and monotonicity of a function based on its graphs
• Transformations of X Horizontal transformation includes translation
  • F(x+c) translates to the left
  • F(x-c) translates to the right
• Horizontal stretching and shrinking
  • Divide x-coordinate by c for y = f(cx)
• Reflections about the y axis for y = f(-x)

Transformations of Y:

• Vertical Translation
• Shifting the function y = f(x) by c units upward or downward of y = f(x) + c and y = f(x) - c
• Vertical Stretching or Shrinking
• Multiplying the y coordinate for the function with y = cf(|x|)
• Reflecting bout the x axis of the function y= -f(x) can be obtained by reflecting the graph y = f(x) about the x axis.