### Discrete Mathematical Structures - Sets and Subsets

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*Definitions:***Set**: A set is a collection of well-defined objects (elements/members). The elements of the set are said to belong to (or be contained in) the set.

A set may be itself be an
element of some other set.

A set can be a set of sets
of sets and so on.

Sets will be denoted be
capital letters A, B, C, ...

Elements will be denoted by
lower case letters a, b, c, .... x, y,
z.

There are five ways to
describe a set.

1. Describe a set by
describing the properties of the members of the set.

2. Describe a set by listing
its elements.

3. Describe a set by its
characteristic functions

4. Describe a set by a
recursive formula.

5. Describe a set by an
operation (such as union, intersection, complement etc.,) on some other

sets.

**Venn diagrams:**Venn diagrams are used to visualise various properties of the set operations.

The

*Universal Set*__is represented by a large rectangular area.__

*Distributive and DeMorgan’s Laws*

__can be established from Venn diagrams.__

*Distributive laws*

AÈ(BÇC) = (AÈB) Ç (AÈC)

*Exercises:***1. Let A = {1, 2, 3, 4, 5}. Which of the following sets are equal to A?**

a.
{4, 1, 2, 3, 5}

b.
{2, 3, 4}

c.
{1, 2, 3, 4, 5,
6}

d.
{x | x is an
integer and x

^{2}£ 25}
e.
{x | x is a
positive integer and x £ 5}

f.
{x | x is a
positive rational number and x £ 5}

**etc...**

*Sequences*
A

**sequence**is a list of objects arranged in a definite order; a first element, second element, third element, and so on. It is classified as**finite**sequence and**infinite**sequence. The elements may all be**different**, or some may be**repeated**.*Examples:*

1. The sequence 1, 0, 0, 1, 0, 1, 0, 0, 1, 1 is a finite
sequence with repeated items.

2. The sequence 1, 3, 5, 7, 9, 11, 13, 15 is a finite
sequence with different items.

3. The sequence 3, 8, 13, 18, … is an infinite sequence.

**etc...**

*Characteristic Functions*
If A is a subset of a universal set

*U*, the**characteristic function**f_{A}of A is defined for each x Î*U*as:
f

_{A}(x) = 1, if x Î A
=
0, if x Ï A.

**etc...**

__Strings__

Given a set A, we can construct the set A* consisting
of all finite sequences of elements of A. Often, the set A is not a set of
numbers, but some set of symbols and Set A is called an

**alphabet**, and the finite sequences in A* are called**words**or**strings**from A. The empty sequence or empty string contains no symbols and denoted as Ù.*Example:*

**etc...**

__Regular Expressions__
A regular expression (over A) is a string constructed
from the elements of A and the symbols (, ), Ú, *, Ù, according to the following definition.

RE1. The symbol Ù is a regular expression.

RE2. If x Î A, the symbol x is a regular expression.

RE3. If a and b are regular expressions, then the expression ab is regular.

RE4. If a and b are regular expressions, then the expression (aÚb) is regular.

RE5. If a is a regular expression, then the expression (a)* is regular.

*Examples:*

1. 0*(0Ú1)*

**etc...**

*Matrices*
A

**matrix**is a rectangular array of numbers arranged in m horizontal**rows**and n vertical**columns**.
a

_{11 }a_{12 … }a_{1n}
a

_{21 }a_{22 … }a_{2n}
A = . . . .

. . . .

a

_{m1 }a_{m2 … }a_{mn}
A = [ a

_{ij}].

**etc...***Boolean matrix*

A

**Boolean matrix**is an m x n matrix whose entries are either zero or one.*Boolean matrix operations*

Let A and B be m x n Boolean matrices.

**etc...**
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