MMW-Lesson2-SETS

SETS

A set is a collection of well-defined objects, ideas or numbers. The groups are called sets for as long as the objects in the group share a characteristic and are thus, well defined.

The objects, ideas or numbers in a set are called elements (∈) of the set. To describe a set, we use “{ }”, and use capital letters to represent it.

Example of Set

  • The books in the shelves in a library.
  • The set of natural numbers N = {1, 2, 3, …}
  • The integers Z = {…, -3, -2, -1, 0,1, 2, 3, …}
  • The set of rainbow colors R = {Red, Orange, Yellow, Green, Violet, Indigo, Blue}

Basic Terms in Set

Ellipses (…) - the three (3) dots in enumerating the elements of the set. When used before/after the first/last element, it denotes infinity. It suggested that there were elements between two elements when it was used between them.

Example of Ellipses

  • A = {1,2, 3, 4, …}
  • B = {4, 8, 12, 16, …}
  • C = {2, 4, 6, …, 30}
  • D = {a, b, c, d, …, z}

Finite VS Infinite Set

A finite set that contains elements that can be counted and terminates at a certain object, idea or number.

An infinite set is a set that contain endless number of elements.

Examples:

  • Set of US states
  • Set of positive numbers
  • Q = {12, 24, 26, …}
  • Rainbow colors
  • Set of first year students in DHVSU
  • G = {…, -8, -4, 0}

Empty Set and Singleton

Empty Set / Null Set - a set with no element denoted by ∅ or “{ }”.

Singleton - a set with only one member.

Equal Sets VS Equivalent Sets

Equal Sets - two or more sets are equal if they contain the same elements.

Equivalent Sets - if the number of elements in two or more sets is the same, they are equivalent.

Examples:

  • {3, 8, 9} and {8, 3, 9}
  • {6, 7, 7, 7} and {6, 7}
  • {🐶, 🐰, 🐼, 🦁} and {🍇, 🍍, 🍋, 🍉}
  • {a, e, i, o, u} and {b, c, d, f, g}
  • { } and ∅
  • {∅} and {5}

Cardinality of Set

The number of distinct elements that belongs to a finite set. Also called as the cardinal number of the set. Denoted by n(A) or |A|.

Universal Set - a set that contains all the elements considered in a particular situation. Denoted by the capital letter “U”.

Subset - is a set whose element belongs to another set or universal set. Denoted by “⊆”.

Let: U = {0,1,2,3,4,5,6,7,8}

  • A = {1,3,4,7}
  • B = {4,7}
  • C = {1,2,7}
  • D = {0,1,2,3,4,5,6,7,8}

Examples:

  • A ⊆ U
  • B ⊆ U
  • C ⊆ U
  • D ⊆ U
  • U ⊆ D

Proper Subset - a subset that Is not equal to the original set. Denoted by “⊂”.

Let: A = {1, 2, 3, 4, 5}

  • B = {1, 2, 3} 𝐵 ⊂ 𝐴
  • C = {3,4} 𝐶 ⊂ 𝐴
  • D = {1} 𝐷 ⊂ 𝐴

Ways of Presenting Sets

There are three main ways in presenting a set

  • Descriptive Form
  • Roster Method
  • Set-Builder Notation

Descriptive Form - defining the rules of which the generates or defines the members.

Example:

  • Set of Rainbow Colors
  • Set of even numbers greater that 3 but less than 100
  • Nothing else belongs to F
  • 8 is the only element of Y

Roster Method - also called List Notation. A way of presenting sets by listing all its member separated by commas and enclosed in a pair of braces.

Example:

  • Set of Rainbow Colors

R = {Red, Orange, Yellow, Green, Violet, Indigo, Blue}

  • Set of even numbers greater than 3 but less than 15

E = {4, 6, 8, 10, 12, 14}

  • Nothing else belongs to F

F = { }

  • 8 is the only element of Y

Y = {8}

Set-Builder Notation - also Predicate Notation/Rule Method. Way of Presenting sets by stating the characteristics or properties of its elements. It has a property that the members of the set share.

Example:

  • Set of Rainbow Colors

{x| x is a rainbow color}

  • Set of even numbers greater than 3 but less than 15

{p | p is an even number greater than 3 and less than 15}

  • Set of English Vowels

{v | v is an English vowel}

  • Student in DHVSU

{d | d is a DHVSU student}

Operations on Sets

There are 3 most common operations on sets

  • Union
  • Intersection
  • Complementation

Finding the elements for a set is what we are performing on the operations on sets.

Union of Sets “⋃” - an operation for two or more sets in which another set is formed by combining all the elements. Sometimes depicted by the word “or".

Let A = {1,2,3,4,5}, B = {2,4,6,8,10}, C = {3,6,9,12,15}, D = {4,8,16,20,24} and E = {5,10,15,20,25}

Find the following:

  • B U C =
  • A U (B U D) =
  • (A U E) U C =
  • B or E =

Intersection of Sets “⋂” - an operation for two or more sets in which another set is formed by listing the common elements.

Let A = {1,2,3,4,5}, B = {2,4,6,8,10}, C = {3,6,9,12,15}, D = {4,8,16,20,24} and E = {5,10,15,20,25}

Find the following:

  • B ꓵ C =
  • A ꓵ B ꓵ C =
  • A ꓵ D =
  • A ꓵ B ꓵ C ꓵ D ꓵ E =
  • A ꓵ (B ꓵ D) =

Compliment of a Set - an operation on a set that must be performed in reference to universal set. Denoted by A’. All elements that “does not belong to” A but on U.

Let U = {0,1,2,3,4,5,6,7,8,9}, A = {1,3,5,7,9}, and B = {2,3,5,7}

Find the following:

  • A’ =
  • B’ =
  • (A ꓵ B)’ =
  • A’ U B’ =
  • (A U B)’ =