Basic Concepts of Set Theory

2.1 Basic Concepts

2.2 Subsets

2.3 Venn Diagrams

2.4 Set Operations & Venn Diagrams (3 sets)

2.5 Surveys and Applications

What is a Set ?

A set is a collection of objects, whose contents can be clearly determined. (Nothing Debatable Allowed)

The objects of the set are called the elements, or members of the set.

Capital letters are used to represent (or name) sets. i.e, A, B, C S, T, etc.

Set is denoted by writing all its elements in curly brackets

For Example B={a, b, c} is a set of with 3 objects denoted by a, b, c.

Notations

Sets are commonly given names (capital letters).

For example A = {1, 2, 3, 4}

The set containing no elements is called the

empty set (null set) and denoted by { } or .

{1,2,3,4}a

The notation ∈ means that a given object is an

element of the set 2 {1,2,3,4}

The notation ∈ means that a given object is not

an element of the set For example

Methods for Describing Sets

Word Description The set of even counting numbers less than 10

The Roster Method, or (listing method)

{2, 4, 6, 8}

Set-Builder notation {x | x is an even counting number less than 10}

| , : are read as “such that”

Sets of Numbers

Natural Numbers (counting) {1, 2, 3, 4, …}, Natural numbers are represented by N Whole Numbers {0, 1, 2, 3, 4, …} Integers {…,–3, –2, –1, 0, 1, 2, 3, …} Rational Numbers All those numbers which can be expressed as ratio of two integers. ½, 1/3 etc These may be written as a terminating decimal, like 0.5, or a repeating decimal like 0.333… Irrational Numbers An irrational number is that cannot be expressed as a ratio of integers. These numbers have decimal expansions that neither terminate nor repeats For Example: Pi=3.1415…………………………. Real Numbers Real Numbers are all those numbers that can be expressed on a real number line.(All of above)

It is the biggest set of the number system, which includes all above five categories.

Graphical view of Number System

Both rational and irrational numbers are a subsets of Real numbers

Inequalities & Set Representation

Inequality Symbol

Word Description Set Builder Notation Roster Method

𝑥 < 4 𝑥 is less than 4

{𝑥|𝑥 ∈ 𝑵 𝑎𝑛𝑑 𝑥 < 4} {1,2,3}

𝑥 ≤ 4 𝑥 is less than or equal to 4

{𝑥|𝑥 ∈ 𝑵 𝑎𝑛𝑑 𝑥 ≤ 4} {1,2,3,4}

𝑥 > 4 𝑥 is greater than 4

{𝑥|𝑥 ∈ 𝑵 𝑎𝑛𝑑 𝑥 > 4} {5,6,7,8, …}

𝑥 ≥ 4 𝑥 is greater than or equal to 4

{𝑥|𝑥 ∈ 𝑵 𝑎𝑛𝑑 𝑥 ≥ 4} {4,5,6,7, …}

4 < 𝑥 < 8 𝑥 is greater than 4 and less than 8

{𝑥|𝑥 ∈ 𝑵 𝑎𝑛𝑑 4 < 𝑥 < 8} {5,6,7}

4 ≤ 𝑥 ≤ 8 𝑥 is greater than or equal to 4 and less than or equal to 8

{𝑥|𝑥 ∈ 𝑵 𝑎𝑛𝑑 4 ≤ 𝑥 ≤ 8} {4,5,6,7,8}

4 ≤ 𝑥 < 8 𝑥 is greater than or equal to 4 and less than 8

{𝑥|𝑥 ∈ 𝑵 𝑎𝑛𝑑 4 ≤ 𝑥 < 8}

{4,5,6,7}

4 < 𝑥 ≤ 8 𝑥 is greater than 4 and less than or equal to 8

{𝑥|𝑥 ∈ 𝑵 𝑎𝑛𝑑 4 < 𝑥 ≤ 8}

{5,6,7,8}

Inequality Symbol

Word Description Set Builder Notation Roster Method

𝑥 < 4 𝑥 is less than 4

{𝑥|𝑥 ∈ 𝑵 𝑎𝑛𝑑 𝑥 < 4} {1,2,3}

𝑥 ≤ 4 𝑥 is less than or equal to 4

{𝑥|𝑥 ∈ 𝑵 𝑎𝑛𝑑 𝑥 ≤ 4} {1,2,3,4}

𝑥 > 4 𝑥 is greater than 4

{𝑥|𝑥 ∈ 𝑵 𝑎𝑛𝑑 𝑥 > 4} {5,6,7,8, …}

𝑥 ≥ 4 𝑥 is greater than or equal to 4

{𝑥|𝑥 ∈ 𝑵 𝑎𝑛𝑑 𝑥 ≥ 4} {4,5,6,7, …}

4 < 𝑥 < 8 𝑥 is greater than 4 and less than 8

{𝑥|𝑥 ∈ 𝑵 𝑎𝑛𝑑 4 < 𝑥 < 8} {5,6,7}

4 ≤ 𝑥 ≤ 8 𝑥 is greater than or equal to 4 and less than or equal to 8

{𝑥|𝑥 ∈ 𝑵 𝑎𝑛𝑑 4 ≤ 𝑥 ≤ 8} {4,5,6,7,8}

4 ≤ 𝑥 < 8 𝑥 is greater than or equal to 4 and less than 8

{𝑥|𝑥 ∈ 𝑵 𝑎𝑛𝑑 4 ≤ 𝑥 < 8}

{4,5,6,7}

4 < 𝑥 ≤ 8 𝑥 is greater than 4 and less than or equal to 8

{𝑥|𝑥 ∈ 𝑵 𝑎𝑛𝑑 4 < 𝑥 ≤ 8}

{5,6,7,8}

Inequalities & Set Representation

Cardinal Number

The number of elements in a set is called the cardinal number, or cardinality of the set. The symbol n(A), read “n of A,” represents the cardinal number of set A. When finding cardinal number, don’t count repeated elements.

Example: Find Cardinal Number

Find the cardinal number of each set. a) K = {2, 4, 6, 6, 8, 10, 12} b) M = {2} c) ∅

Solution a) n(K) = 6 b) n(M) = 1 c) n(∅)=0

Equal Sets Set A is equal to set B provided the following two conditions are met: 1. Every element of A is an element of B, & 2. Every element of B is an element of A.

Then we write equal sets as A=B Order of elements or possible repetition of elements does not matter, must be disregarded.

Equivalent Sets

Two sets are Equivalent if they have the same number of elements. If two sets are equal then they must be

equivalent. However if two sets are equivalent, they are

not necessarily equal.

Example State whether the sets in each pair are equal or equivalent i) A={a, b, c, d} and B= {b, c, d, a} ii) A={2, 4, 6,8,10} and B={i, j, k, l,m}

Solution i) Equal, & (Every element of A is in B, and every element of B is in A) Equivalent (Both sets have same number of elements)

ii) Not equal, (Elements of A are NOT in B and …. ) But Equivalent (Both sets have same number of elements)

• We will do examples similar to questions assigned to you as Home Work Problems

• Make sure to understand these examples, before you attempt HW and Quiz.

• Exam Problems will be similar to HWs

Examples-1: Which collections are not well defined, and therefore not sets

Question: “The collection of days in a week” Is this Collection a Set ? Is it well defined ? Answer: Well Defined; so it’s a SET

Question: “The collection of your favorite days in a week” Is this Collection a Set ? Is it well defined ? Answer: Not Well Defined; so it’s NOT a SET

Examples-2: Write word description of each set, more than one correct answers possible

Question: {April, August} This is given in Roster Method Goal is to describe what you see in words Answer: “The set of months that begin with letter A”

Question: {9,10,11,12,…,25} Given in a roster method, lets write word description Answer: “The set of Natural numbers between 9 and 25 including both 9 and 25”

Examples-3: Write these sets in roster method

Question: The set of months that have exactly 30 days

This is given in word description

Goal is to write it in roster method

Answer: {April, June, September, November}

Question: {𝑥|𝑥 ∈ 𝑵 𝒂𝒏𝒅 𝟔 < 𝒙 ≤ 𝟏𝟎} Given in set builder notation

Answer: {7,8,9,10}

Examples-4: Write these sets in roster method

Question: {𝑥|𝑥 + 5 = 7} Given in set builder notation

Answer: Solve x+5=7 so {2}

Question: 𝑥 𝑥 < 0 𝒂𝒏𝒅 𝒙 > 5} Given in set builder notation

Answer: {∅} Because there is no x which is less than zero and bigger than 5

Examples-5: Find the cardinal number of a given set (The number of elements in a given set )

Question: A={17,19,21,23,25} Find cardinal number Number of elements is five, so

Answer: The Cardinal Number is 5, or n(A)=5

Question: C= {𝑥|𝑥 𝑖𝑠 𝑎 𝑑𝑎𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑤𝑒𝑒𝑘 𝑡𝑎ℎ𝑡 𝑏𝑒𝑔𝑖𝑛𝑠 𝑤𝑖𝑡ℎ 𝑙𝑒𝑡𝑒𝑡𝑟 𝐴}

Lets count the number of elements in this set There is no such day, so its an empty set Answer: The Cardinal number is 0, or n(A)=0

Examples-6: Find if the given sets are EQUAL or Equivalent

Question: A={1,1,1,2,2,3,4}, B={4,3,2,1} Answer: A & B are equal A & B are Equivalent as well Question: 𝐴 = {𝑥|𝑥 ∈ 𝑵 𝑎𝑛𝑑 6 ≤ 𝑥 < 10} 𝐵 = {𝑥|𝑥 ∈ 𝑵 𝑎𝑛𝑑 9 < 𝑥 ≤ 13}

Lets write them using Roster method 𝐴 = {6,7,8,9} and B= {10,11,12,13} Answer: A & B are NOT equal A & B are Equivalent