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Set Theory in Discrete Mathematics

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Read on to learn about set theory in discrete mathematics.

Read on to learn about set theory in discrete mathematics.

Set Theory in Discrete Mathematics

Every branch of mathematics uses the idea of a set. Georg Cantor, a German mathematician and philosopher, developed a theory of abstract sets of entities and turned it into a field of mathematics between 1874 and 1897. His research into a few real-world issues involving specific kinds of infinite sets of real numbers resulted in the creation of this theory.

Definition of a Set and Elements

Any clearly defined group of things, known as the set's members or elements, can be considered a set.

Typically, sets are denoted by capital letters, such as A, B, X, Y, etc., while components of sets are denoted by lowercase letters, such as a, b, x, y, etc. Set can also be referred to as a "class," "collection," or "family."

Following is how membership in a set is indicated.

  • a ∈ S denotes that a belongs to a set S
  • a, b ∈ S denotes that a and b belong to a set S Here ∈ is the symbol meaning “is an element of.”

How Can We Represent Sets?

There are really just two methods for specifying a certain set. List the group's members, if possible, with commas separating them and enclosing them in braces. The characteristics of the items in the set are stated in a second method. Examples of these two methods are as follows:

Method No.1

A = {1, 2, 9, 0}

The set A consists of the numbers 1, 2, 9, 0.

Method No.2

B= {x | x is an odd integer, x > 0}

B is the set of x such that x is an odd integer and x is greater than 0. Signifies the positive integer-containing set B. Keep in mind that a letter, often x, is used to identify a typical member of the set, and that the comma and vertical line | are both read as "and."

Examples:

  • We cannot list all the elements of the above set B although frequently we specify the set by B = {2, 4, 6...} where we assume that everyone knows what we mean. Observe that 8 ∈ B, but 3 is not belongs to B.
  • The set A above can also be written as A = {x | x is an odd positive integer, x < 10}

What Are Subsets?

Assume that each element in set A is also an element in set B, or that a set A implies a set B. So, A is considered to as a subset of B. We can also state that B contains A or that A is contained in B. This association is noted as,

A ⊆ B or B ⊇ A

Equal Sets

If two sets contain the same items, or, to say it another way, if one set contains the other, then the sets are equal. Which is:

A = B if and only if A ⊆ B and B ⊆ A

Example of Subsets:

Let us consider the three sets

A = {1, 2, 3, 6} B = {2, 4, 6, 8,} C = {1, 3}

C ⊆ A because contain in A. But B is not a subset of A because it does not contain in A.

Theorem (Subsets):

Let A, B, C be any sets.

Then:

(i) A ⊆ A

(ii) If A ⊆ B and B ⊆ A, then A = B

(iii) If A ⊆ B and B ⊆ C, then A ⊆ C

Special Symbols of Sets:

We use special symbols for some sets because they will appear in the text quite frequently. Some examples of these symbols are:

N = the set of positive integers or natural numbers: 1, 2, 3...

Z = the set of all integers: ..., −2, −1, 0, 1, 2...

Q = the set of rational numbers R = the set of real numbers C = the set of complex numbers

Result:

N ⊆ Z ⊆ Q ⊆R ⊆ C.

Universal Set

All sets under consideration in any application of set theory are taken to belong to some fixed huge set called the universal set, which we designate by U.

Example of Universal Set:

Given a universal set U and a property P, there may not be any elements of U which have property P. For example, the following set has no elements:

S = {x | x is a positive integer, x^2 = 3}

There is just one empty set, also known as the null set, which is a set without any elements. To put it another way, if S and T are both empty, then S = T because they both have the exact same items, namely none. Every other set is also thought of as a subset of the empty set. As a result, we can formalize the straightforward conclusion that follows.

Theorem (Universal Set):

Statement:

For any set A,

we have ∅ ⊆ A ⊆ U

Disjoint Sets

Definition:

If two sets A and B don't overlap by any elements, they are said to be disjoint.

Example of Disjoint Sets:

Let us consider the three sets like,

A = {1, 3, 5} B = {6, 8, 9} C = {6, 5, 4}

The set A and B are disjoint, but B and C are not disjoint because one element 6 is common in them.

Applications of Set Theory

Applications of Set Theory in Science

Applications of set theory are most frequently used in science and mathematics fields like biology, chemistry, and physics as well as in computer and electrical engineering. These applications range from forming logical foundations for geometry, calculus, and topology to creating algebra revolving around field, rings, and groups.

Applications of Set Theory in Real Life

In the kitchen, tools are arranged such that plates are kept apart from spoons. This is another example of a set in real life. Another illustration is how advanced phones like the Galaxy Duos, Lumia, and others are separated from basic mobiles when we visit mobile stores.

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

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