Sets | Mathematics for ACT PDF Download

Introduction

Sets in mathematics, are simply a collection of distinct objects forming a group. A set can have any group of items, be it a collection of numbers, days of a week, types of vehicles, and so on. Every item in the set is called an element of the set. Curly brackets are used while writing a set. A very simple example of a set would be like this. Set A = {1,2,3,4,5}. There are various notations to represent elements of a set. Sets are usually represented using a roster form or a set builder form.

Sets Definition

In mathematics, a set is a well-defined collection of objects. Sets are named and represented using a capital letter. In the set theory, the elements that a set comprises can be any kind of thing: people, letters of the alphabet, numbers, shapes, variables, etc.

Sets in Maths Examples

We know that a collection of even natural numbers less than 10 is defined, whereas collection of intelligent students in a class is not defined. Thus, collection of even natural numbers less than 10 can be represented in the form of a set, A = {2, 4, 6, 8}. Let us use this example to understand the basic terminology associated with sets in math.

Elements of a Set

The items present in a set are called either elements or members of a set. The elements of a set are enclosed in curly brackets separated by commas. To denote that an element is contained in a set, the symbol '∈' is used. In the above example, 2 ∈ A. If an element is not a member of a set, then it is denoted using the symbol '∉'. Here, 3 ∉ A.

Cardinal Number of a Set

The cardinal number, cardinality, or order of a set denotes the total number of elements in the set. For natural even numbers less than 10, n(A) = 4. Sets are defined as a collection of unique elements. One important condition to define a set is that all the elements of a set should be related to each other and share a common property. For example, if we define a set with the elements as the names of months in a year, then we can say that all the elements of the set are the months of the year.

Representation of Sets

There are different set notations used for the representation of sets. They differ in the way in which the elements are listed. The three set notations used for representing sets are:

• Semantic form
• Roster form
• Set builder form

Semantic Form 
The semantic notation describes a statement to show what are the elements of a set. For example, Set A is the list of the first five odd numbers.

Roster Form
The most common form used to represent sets is the roster notation in which the elements of the sets are enclosed in curly brackets separated by commas. For example, Set B = {2,4,6,8,10}, which is the collection of the first five even numbers. In a roster form, the order of the elements of the set does not matter, for example, the set of the first five even numbers can also be defined as {2,6,8,10,4}. Also, if there is an endless list of elements in a set, then they are defined using a series of dots at the end of the last element. For example, infinite sets are represented as, X = {1, 2, 3, 4, 5 ...}, where X is the set of natural numbers. To sum up the notation of the roster form, please take a look at the examples below.
Finite Roster Notation of Sets : Set A = {1, 2, 3, 4, 5} (The first five natural numbers)
Infinite Roster Notation of Sets : Set B = {5, 10, 15, 20 ....} (The multiples of 5)

Set Builder Form

The set builder notation has a certain rule or a statement that specifically describes the common feature of all the elements of a set. The set builder form uses a vertical bar in its representation, with a text describing the character of the elements of the set. For example, A = { k | k is an even number, k ≤ 20}. The statement says, all the elements of set A are even numbers that are less than or equal to 20. Sometimes a ":" is used in the place of the "|".

Visual Representation of Sets Using Venn Diagram

Venn Diagram is a pictorial representation of sets, with each set represented as a circle. The elements of a set are present inside the circles. Sometimes a rectangle encloses the circles, which represents the universal set. The Venn diagram represents how the given sets are related to each other.

Sets Symbols

Set symbols are used to define the elements of a given set. The following table shows some of these symbols and their meaning.

Types of Sets

Sets are classified into different types. Some of these are singleton, finite, infinite, empty, etc.

• Singleton Sets
  A set that has only one element is called a singleton set or also called a unit set. Example, Set A = { k | k is an integer between 3 and 5} which is A = {4}.
• Finite Sets
  As the name implies, a set with a finite or countable number of elements is called a finite set. Example, Set B = {k | k is a prime number less than 20}, which is B = {2,3,5,7,11,13,17,19}
• Infinite Sets 
  A set with an infinite number of elements is called an infinite set. Example: Set C = {Multiples of 3}.
• Empty or Null Sets 
  A set that does not contain any element is called an empty set or a null set. An empty set is denoted using the symbol '∅'. It is read as 'phi'. Example: Set X = {}.
• Equal Sets 
  If two sets have the same elements in them, then they are called equal sets. Example: A = {1,2,3} and B = {1,2,3}. Here, set A and set B are equal sets. This can be represented as A = B.
• Unequal Sets 
  If two sets have at least one element that is different, then they are unequal sets. Example: A = {1,2,3} and B = {2,3,4}. Here, set A and set B are unequal sets. This can be represented as A ≠ B.
• Equivalent Sets 
  Two sets are said to be equivalent sets when they have the same number of elements, though the elements are different. Example: A = {1,2,3,4} and B = {a,b,c,d}. Here, set A and set B are equivalent sets since n(A) = n(B)
• Overlapping Sets 
  Two sets are said to be overlapping if at least one element from set A is present in set B. Example: A = {2,4,6} B = {4,8,10}. Here, element 4 is present in set A as well as in set B. Therefore, A and B are overlapping sets.
• Disjoint Sets 
  Two sets are disjoint sets if there are no common elements in both sets. Example: A = {1,2,3,4} B = {5,6,7,8}. Here, set A and set B are disjoint sets.
• Subset and Superset
  For two sets A and B, if every element in set A is present in set B, then set A is a subset of set B(A ⊆ B) and B is the superset of set A(B ⊇ A).
  Example: A = {1,2,3} B = {1,2,3,4,5,6}
  A ⊆ B, since all the elements in set A are present in set B.
  B ⊇ A denotes that set B is the superset of set A.
• Universal Set
  A universal set is the collection of all the elements in regard to a particular subject. The universal set is denoted by the letter 'U'.
  Example: Let U = {The list of all road transport vehicles}. Here, a set of cars is a subset for this universal set, the set of cycles, trains are all subsets of this universal set.
• Power Sets
  Power set is the set of all subsets that a set could contain.
  Example: Set A = {1,2,3}. Power set of A is = {{∅}, {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3}}.

Sets Formulas

Sets find their application in the field of algebra, statistics, and probability. There are some important set formulas as listed below. For any two overlapping sets A and B,

• n(A U B) = n(A) + n(B) - n(A ∩ B)
• n (A ∩ B) = n(A) + n(B) - n(A U B)
• n(A) = n(A U B) + n(A ∩ B) - n(B)
• n(B) = n(A U B) + n(A ∩ B) - n(A)
• n(A - B) = n(A U B) - n(B)
• n(A - B) = n(A) - n(A ∩ B)

For any two sets A and B that are disjoint,

• n(A U B) = n(A) + n(B)
• A ∩ B = ∅
• n(A - B) = n(A)

Properties of Sets

Similar to numbers, sets also have properties like associative property, commutative property, and so on. There are six important properties of sets. Given, three sets A, B, and C, the properties for these sets are as follows.

Operations on Sets

Some important operations on sets include union, intersection, difference, the complement of a set, and the cartesian product of a set. A brief explanation of operations on sets is as follows.

• Union of Sets
  Union of sets, which is denoted as A U B, lists the elements in set A and set B or the elements in both set A and set B. For example, {1, 3} ∪ {1, 4} = {1, 3, 4}
• Intersection of Sets
  The intersection of sets which is denoted by A ∩ B lists the elements that are common to both set A and set B. For example, {1, 2} ∩ {2, 4} = {2}
• Set Difference
  Set difference which is denoted by A - B, lists the elements in set A that are not present in set B. For example, A = {2, 3, 4} and B = {4, 5, 6}. A - B = {2, 3}.
• Set Complement
  Set complement which is denoted by A', is the set of all elements in the universal set that are not present in set A. In other words, A' is denoted as U - A, which is the difference in the elements of the universal set and set A.
• Cartesian Product of Sets
  The cartesian product of two sets which is denoted by A × B, is the product of two non-empty sets, wherein ordered pairs of elements are obtained. For example, {1, 3} × {1, 3} = {(1, 1), (1, 3), (3, 1), (3, 3)}.
  Operations of Sets and Venn Diagrams
  

Example on Sets

Example: Find the elements of the sets represented as follows and write the cardinal number of each set. a) Set A is the first 8 multiples of 7 b) Set B = {a,e,i,o,u} c) Set C = {x | x are even numbers between 20 and 40}
Solution:
a) Set A = {7,14,21,28,35,42,49,56}. These are the first 8 multiples of 7.
Since there are 8 elements in the set, cardinal number n (A) = 8
b) Set B = {a,e,i,o,u}. There are five elements in the set,
Therefore, the cardinal number of set B, n(B) = 5.
c) Set C = {22,24,26,28,30,32,34,36,38}. These are the even numbers between 20 and 40, which make up the elements of the set C.
Therefore, the cardinal number of set C, n(C) = 9.

Example 2: If Set A = {a, b, c}, Set B = {a, b, c, p, q, r}, U = {a, b, c, d, p, q, r, s}, find the following using sets formulas, a) A U B b) A ∩ B c) A' d) Is A ⊆ B? (Here 'U' is the universal set).