MTH100 Handouts Updated by Team Aiza

Sets and Numbering Systems

Topic 1

The study of Mathematics begins with a study of sets and the development of the numbering systems. Every mathematical system can be represented as a “set”; therefore, it is important for us to understand the definitions, notations and properties of “sets”

Definition: A set is an unordered collection of distinct objects. Objects in the collection are called elements of the set.

Examples: o The collection of persons living in Lahore is a set.

 Each person living in Lahore is an element of the set.

o The collection of all towns in the Punjab province is a set.

 Each town in Punjab is an element of the set.

o The collection of all quadrupeds is a set.

 Each quadruped is an element of the set.

o The collection of all four-legged dogs is a set.

 Each four-legged dog is an element of the set.

o The collection of counting numbers is a set.

 Each counting number is an element of the set.

o The collection of pencils in your bag is a set.

 Each pencil in your bag is an element of the set.

Notation: Sets are usually designated with capital letters. Elements of a set are usually designated with lower case letters. o D is the set of all four legged dogs. o An individual dog might then be designated by d.

The roster method of specifying a set consists of surrounding the collection of elements with braces. For example the set of counting numbers from 1 to 5 would be written as {1, 2, 3, 4, 5}.

Set builder notation has the general form {variable | descriptive statement }.

The vertical bar (in set builder notation) is always read as “such that”. Set builder notation is frequently used when the roster method is either inappropriate or inadequate.

For example, {x | x < 6 and x is a counting number} is the set of all counting numbers less than 6. Note this is the same set as {1,2,3,4,5}.

Other Notation: If x is an element of the set A, we write this as x  A. x

 A

means x is not an element of A.

If A = {3, 17, 2 } then 3  A, 17  A, 2  A and 5  A. If A = { x | x is a prime number } then 5  A, and 6  A.

Definition: The set with no elements is called the empty set or the null set and is designated with the symbol

Definition: The universal set is the set of all things pertinent to a given discussion and is designated by the symbol U

For example, when dealing with all the students enrolled at the Virtual University, the Universal set would be U = {all students at the Virtual University} Some sets living in this universal set are: A = {all Computer Technology students} B = {first year students} C = {second year students}

Definition: The set A is a subset of the set B, denoted A  B, if every element of A is an element of B. If A is a subset of B and B contains elements which are not in A, then A is a proper subset of B. It is denoted by A  B.

If A is not a subset of B we write A  B to designate that relationship.

Definition: Two sets A and B are equal if A  B and B  A. If two sets A and B are equal we write A = B to designate that relationship. In other words, two sets, A and B, are equal if they contain the same elements

Definition: The intersection of two sets A and B is the set containing those elements which are elements of A and elements of B. We write A  B to denote A Intersection B.

 B = {3, 6} Definition: The union of two sets A and B is the set containing those elements which are elements of A or elements of B. We write A  B to denote A Union B.

 B = {1, 2, 3, 4, 5,

6}. • Algebraic Properties of Sets:

o Union and intersection are commutative operations. In other words, A

 B = B

 A and A ∩ B = B ∩ A o Union and intersection are associative operations. In other words,

(A ·  B) ·  C = A ·  (B ·  C) and (A ∩ B) ∩ C = B ∩ (A ∩ C)

Union and Intersection are distributive with respect to each other. In other words

A ∩ ( B

 C )= (A ∩ B)

 (A ∩ C) and A

 ( B ∩ C )= (A

 B) ∩ (A

 C) • A few other elementary

properties of intersection and union.

A ·  =A, A ∩ ·  , A ·  A = A, A ∩ A = A. •

Numbering Systems:

Counting numbers are called Natural numbers and the set of Natural numbers is denoted by N = {1,2,3…} Integers are Natural numbers, their opposites and zero. The set of integers is denoted by Z = {…3, -2, -1, 0,1,2,3…} Rational numbers such as 2/3, -31/2 0.3333, are numbers that can be written as a ratio of two integers. The set of rational numbers is denoted by Q. This set includes o Repeating decimals, terminating decimals and fractions o Integers are also rational numbers since every integer a can be written as a fraction a/1 Irrational numbers are numbers that can’t be written as fractions. o 3. 45455455545555… has a pattern but doesn’t repeat. It isn’t rational. It can’t be written like a fraction. o Square root of 2,  (Pi) and e are also irrational. The Union of the set of rational numbers and the set of irrational numbers is the set of Real numbers, denoted by R

N ·  Z ·  Q ·  R

Definition: Cardinality refers to the number of elements in a set o A finite set has a countable number of elements o An infinite set has at least as many elements as the set of natural numbers

Notation: |A| represents the cardinality of Set A