Lesson 1 BASIC PROPERTIES FROM SET THEORY

WHAT IS A SET?

1. A set is a well-defined collection of distinct objects.

· Well-defined means clearly stated or clearly defined.

· Distinct means different or distinguished from others.

· Objects are called ELEMENTS.

· Є(epsilon) is used to denote elements.

2. A set is enclosed in a curly brace{} and objects/elements are separated by a comma.

· … (ellipsis) is used to indicate infinite set or continuity of elements with the same pattern.

3. A set is denoted by a capital letter of the English alphabet.

· Null or empty set is a set with no element. Ф or {}

· Singleton is a set with only one element.

· Pair is a set with two elements.

Assessment: Check Your Understanding on page 5

HOW TO DESCRIBE A SET?

1. Roster Method – listing all the elements of the set exactly once.

Ex. A = {a,e,i,o,u}

2. Set-Builder Method – stating one determining feature of its elements.

Ex. A = {x|x is a vowel in the English alphabet}

· | - is read as “such that”

Assessment: Check Your Understanding on page 5

SUBSETS AND PROPER SETS

1. Proper Subset – if all the elements of set A is obtained in set B but there are elements of set B that is not obtained in set A.

Ex. A = {a,c,e} B = {a,b,c,d,e} AB or BA

2. Subset – parts of a set. It is an element or group of elements that are parts of a certain set. Its notation is. To solve for the number of subsets a set has, use the formula 2ⁿ where n is the number of elements of the set.

Ex. A = {a,c,e} Its subset are {}, {a}, {c}, {e}, {a,c}, {a,e}, {c,e}, {a,c,e}.

· The collection of the subsets of a set is called power set.

REMEMBER: A null set is a subset of all subsets and a set is a subset of itself.

HOW TO COMPARE SETS?

1. Equal Sets (=) – sets with the same elements.

Ex. A = {m,a,l,l} B = {a,l,m} C = {m,a,m,m,a,l}

Remember: elements of a set should be distinct

A = {m,a,l} B = {a,l,m} C = {m,a,l}

In this case, we notice that all sets has elements of a,l and m, therefore A = B, A = C or B = C

1. Equivalent Sets (~) – sets with the same cardinality.

· Cardinality of set means the number of elements in a set.

Ex. A = {m,y,r,a} The cardinality of set A is 4 because it has four elements.

Ex. X = {1,2,3,4,5} Y = {6,5,4,3,2}

The cardinality of set X is 5 and the cardinality of set Y is also 5, therefore X~Y.