Basic Definitions and Concepts of Sets, Relations and Functions

Sets, relations and functions are all part of the set theory. These tools help in carrying out logical and mathematical set operations on mathematical and other real-world entities. Sets help in distinguishing the groups of certain kind of objects. Whereas set operations i.e., relations and functions are the ways we use to connect and to work with sets.

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Sets

A set is the representation of a collection of objects; distinct objects with one or more common properties. Grouping up the objects in a set is an act of distinguishing those objects from the members of another set. We can use the terms – elements or members of a set instead of the term objects.

How to denote Sets?

If ‘A’ is a set and ‘a’ one of its elements then: ‘a ∈ A’ denotes that element ‘a’ belongs to ‘A’ whereas, ‘a ∉ A’ denotes that ‘a’ is not an element of A. Alternatively, we can say that ‘A’ contains ‘a’. A set is usually represented by capital letters and an element of the set by the small letter.

Representation of Sets

There are mainly 3 ways to represent a set:

1. Statement form.
2. Roaster form (tabular method).
3. Set Builder form.

Statement Form

Here, a single statement describes all the elements inside a set. For example:

• V = The set of all vowels in English.

Roaster Form

In this form all the members of the given set are enlisted within a pair of braces { }, separated by commas. For example:

• The set of all even whole numbers between 1 to 10.
  E = {2, 4, 6, 8}

Set Builder Form

Here, a property is stated that must be common to all the elements of that particular set.

• N = { x : x is positive integers between 10 to 20 }

We read the set builder form as ” N is the set of all x such that x is a positive integer between 10 to 20″. Braces{} denote the set while ‘:’ denotes ‘such that’.

Types of Sets

Knowing the type of a set helps in verifying the appropriate set operations applicable to that particular set.

Empty Set

A set with no elements. Empty sets are also called null sets or void sets and are denoted by { } or Φ.

• Ex. {x: x is an integer which is a perfect cube and lies between 2 and 7}.

Equal Sets

Sets with equal elements.

• Example: A = {5, 6, 7} and B = {5, 6, 5, 7, 7}.
• Here, the elements of A and B are equal to each other (5, 6, 7) i. e., A = B

In case of repetition as in B we write B = {5, 6, 7} by ignoring the repetition.

Equivalent Sets

Sets with the equal number of members.

• Ex. A = {3, 6, 8}  and B = {p, q, r}.
• Both A and B having three elements are equivalent sets.
  

Two equal sets are equivalent too but the vice-versa doesn’t always hold true.

Finite and Infinite Sets

Based on the number of elements (finite or infinite) present in the set, the set is either finite or infinite. In case of infinite the set, it is given as:

• N = Numbers divisible by 4 = {4, 8, 12, 16…..}

Singleton Set

A set with a single element. For example, {9}.

Subsets and Super-sets

A set qualifies as the subset of another set if all of its elements are also the elements of that another set. A collection of all the subsets of a given set is a power set. Example, for R = {5, 8} , P(R) = {{5, 8}, {5}, {8}, {}} will be the power set. Now, we can infer that a set with n no. of elements has 2ⁿ no. of subsets or 2ⁿ no.of elements in its power set.

A super-set can be thought of as the parent set that at least contains all the elements of the subset and may or may not contain some extra elements.

• C = The set of all colors and  P = The set of all primary colors then, P ⊂ C; read as P is a “subset of” C or P is “contained in” C.  Or we have:
• C ⊃ P which means C is a “superset of” P or C “contains” P.

Subsets of a Set of Complex Numbers, Source: Wikipedia. 

For the figure given above if we consider

• N = the set of natural numbers
• W = the set of whole numbers.
• I = the set of Integers.
• Rt = the set of rational numbers.
• Re = the set of real numbers.
• C = the set of complex numbers.

We can say that N ⊂ W ⊂ I ⊂ Rt ⊂ Re ⊂ C. Also, going in the reverse order we have C ⊃ Re ⊃ Rt ⊃ I ⊃ W ⊃ N. Here we can call the set of complex numbers as a universal set for real, rational, integers, whole and natural numbers.

Relations and Functions

Relations and functions are the set operations that help to trace the relationship between the elements of two or more distinct sets or between the elements of the same set. But, before we move on to further explore the topic it is important to get the idea about the cartesian product and Venn diagrams.

Cartesian Product

If p ∈ P and q ∈ Q then the set of all ordered pair i. e., (p,q) is called the Cartesian product of P×Q. This means every first element of the ordered pair belongs to the set P and every second element belongs to the set Q.

• P and Q must be non-empty sets.
• P × Q is null if either P or Q is a null set.

Example: X = {1, 3} and Y = {4, 7} then, X × Y = {(1, 4), (1, 7), (3, 4), (3, 7)}.

For above example, the number elements in X, n(X) = 2 (this no. is called the cardinal number) and number elements in Y, n(Y) = 2 consequently, number of ordered pairs in the Cartesian product n(X × Y) = 4. The same rule holds for any Cartesian product.

Venn-Diagrams

Euler – Venn Diagrams make it easy to operate on sets.

In the above example we consider:

• The quadrilateral is a finite universal set which represents a set of ideas.
• Among those ideas we have two subsets which are finite as well:
• Set of Ideas that are the truth, let’s denote them as T and
• Set of Ideas that are the beliefs, let’s denote them as B.
• Their intersection set, knowledge is denoted as  K.

• Here, T ∪ B is the union of these two sets which is the set of ideas that are either T or B or both.
• T ∩ B is the intersection of the two sets which is the set of ideas that are both T and B i. e., K as already mentioned.
• Now, to find no. of elements in T ∪ B we have, n(T ∪ B) = n(T) + n(B) – n(T ∩ B).

Relations and Functions

The relation is the subset of the Cartesian product which contains only some of the ordered pair based on the relationships defined between the first and second elements. The relation is usually denoted by R.

If every element of a set A is related with one and only one element of another set then this kind of relation qualifies as a function. A function is a special case of relation where no two ordered pairs can have the same first element.

This notation f:X→Y denotes that f is a function from X to Y. For x∈X there is  unique y∈Y and his y is represented as y = f(x) which means value of f at x which in turn is the value of y at a specific value of x.

We can represent a function in three ways namely: Algebraic form, Tabular form, and Graphical form.

Domain and Range

• The domain is the set of all first elements of R.
• The range is the set of all second elements of R.

Total no. of relations for n(A×B) are 2^n(A)×n(B)

Types of Functions

Based on the kind of element that the sets involved consists of functions can be:

• Identity function: y=f(x)=x; both range and domain of the function is the same.
• Constant function: y=f(x)=Constant; the range of the function is constant
• Polynomial function: y=f(x)=polynomial for every value of x.
• Rational function: these are y=f(x)=g(x)/h(x) type of function where both g(x) and h(x) are polynomials and h(x)≠0.
• Modulus function: The range of the function is positive plus the set of 0
• Signum function: Range of f is {-1, 0, 1}
• Greatest Integer function: The function assumes the value of the greatest integer. The range is a pure integer value.

Based on the kind of relationship that the elements of  the two sets have with each other there are mainly four types of functions:

• One to one function(Injective): For each element in the domain there is one and only one element in the range.
• Many to one function: When two or more elements from the domain are mapped to the same single elements in the range.
• Onto function(Surjective): When every element of the range has been mapped to an element in the domain.
• One-one and onto function(Bijective): A function which is both one to one and onto function.

Solved Examples on Set Operations

Question. {0} is equal to:

a) 0           b) Φ           c) both a) and b)         d) none.

Ans. d) none; the set {0} has one element 0 and therefore can’t be considered empty and 0 is not a number in {o} but, it is an element of the singleton set so, {0} ≠ 0.

Question. The given function is:

a) Injective        b) Surjective      c) Many to one       d) Bijective

Ans. a) injective each element in X is mapped to a distinct element in Y.

This concludes our discussion on the topic of set operations.