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Sets and Relations

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A set is a collection of well defined objects. Repetition of the elements are immaterial. The number of elements in a set '$A$' is denoted as $|A|$, is called the cardinality of the set, a set with no elements. Relationship between sets can be represented by venn diagrams.
A relation describes certain properties of things and is a set of ordered pairs. It is a subset of ordered pairs drawn from the set of all possible ordered pairs.
$R$ is a relation if 

$R \subseteq {(x, y) |x\ \epsilon \ X,y\ \epsilon Y}$

Here Domain is $X$ and Codomain is $Y$. Collection of well defined objects. The order or repetition of the elements are immaterial. The word 'set' was used by the German Mathematician George Cantor to define a set.

Sets Definition

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A set is a collection of objects. Objects are called elements or members of the set. Elements contained by a set can be unordered.

Example: $X$ = $ \{ a, e, i, o, u \}$ - Vowels in the English alphabets. 

Set may be a collection of objects, things, places, persons, constants, variables, special symbols or characters. 

Different operations can be performed over Sets in discrete Math. These are Union of sets, intersection of sets, and difference of sets and complement of a set.

Set Representation

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There are three ways to represents a set.

1) Descriptive Form: Elements of a set are stated in words.

Examples:

$A$ : Set of first ten even numbers.

$B$ : Set of first five numbers.

$C$ : Set of positive odd numbers.

2) Set Builder form: Common characteristic shared by all the elements of the sets are written in symbolic form.

Example: $C$ : $\{ x ; 1 < X < 5 \}$

3) Tabular form: List the elements of a set separated by commas and enclosed within curly brackets "$\{ \}$".

Examples:

$A$ = $\{ 2, 4, 6, 8, 10 \}$

$B$ = $\{ 1, 2, 3, 4, 5 \}$

$C$ = $\{ 1, 3, 5, ..... \}$

Operations on Sets

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There are various operations are performed over different sets. Set operation may be performed on two or more sets. 

Given below are the four basic operations on sets are shown by venn diagrams. Venn diagrams are used for displaying set relations in a way that is more visual than the algebraic form. A circle is used to represent each subset. The portions of the diagram that are shaded represents the set expression. 
Basic Operations on Sets

1) Intersection of Sets:

Intersection of sets $A$ and $B$, denoted $A \cap B$, read as 'the intersection of $A$ and $B$ is the set containing the elements common to the two sets.

Intersection of sets $A$ and $B$, is the set defined as

$A \cap B$ = $\{ x |x$ is both in $A$ and $B \}$.

Intersection
2) Union of Sets:

Union of sets $A$ and $B$, is the set defined as 

$A \cup B$ = $\{ x\ |x\ \epsilon\ A\ \forall\ x\ \epsilon\ B \}$. 

It is denoted by $A \cup B$, read as 'the union of $A$ and $B$' is defined as the set that consists of all elements belonging to either set $A$ or set $B$.

3) Difference of Sets:

The difference between $A$ and $B$, denoted $A$ - $B$, is the set $\{ x\ |x$ is in $A$ but not in $B \}$.

Difference of Sets
4) Complement of Sets:

For a set $A$, the difference $U$ - $A$, where $U$ is the universe, is called the complement of $A$ and it is denoted by $\bar{A}$ or $A$'.
Thus $\bar{A}$ is the set of everything that is not in $A$.

Complement of Set

Types of Sets

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1) Unit Set: A set that has only one element.

2) Universal Set: A set that contains everything related to the problem.

    Example: In calculus, universal set is real numbers. In complex analysis, universal set is the complex numbers.

3) Finite Set: A set consisting of finite number of elements is called finite set.

4) Disjoint Set: If two sets $A$ and $B$ are such that $A \cap B$ = $\phi$, the two sets are said to be disjoint.

5) Overlapping Sets: If the two sets $A$ and $B$ have some elements common then they are called overlapping sets. That is $A \cap B\ \neq\ \phi$.

    Example: $L$ = $(p, q, r \},\ M$ = $\{q, r, s, t, u\}$, both sets $L$ and $M$ have common elements of $q$ and $r$ so they are called overlapping sets.

Properties of Sets

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Basic properties of set operations are disscused below:

1) Identity Law
   $A \cup \phi$ = $A$ 

2) Domination Law
   $A \cup U$ = $U$ 

3) Commutative Law
   $A \cup B$ = $B \cup A$

4) Associative Law
   $(A \cup B) \cup C$ = $A \cup (B \cup C)$ 

5) Distributive Law
   $A \cup (B \cap C)$ = $(A \cup B) \cap (A \cup C)$ 

6) $A \cap (B - A)$ = $\phi$

7) $B$ = $\bar{A}$ if and only if $A \cup B$ = $\cup$ and $A \cap B$ = $\phi$

8) $A \cap B \subseteq  A$

Subsets

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When all the elements of one set are completely contained in a second set, the first set is a subset of the second.

Example: Consider the sets

$A$ = {Avocado, Blackberry, Strawberry, Kiwifruit, Figs, Peach, Lychee}

$B$ = {Strawberry, Kiwifruit, Figs}

From the above example we see that $B$ is a subset of $A$. Every set is a subset of itself.

Power Sets

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The set of all subsets of a given set $A$ is called the power set of $A$ and is denoted by $P(A)$. A set with $n$ elements has $2^{n}$ subsets.

If $A$ = $\{ 1, 2, 3 \}$, then $P(A)$ = $\{ (\phi), (1), (2), (3), (1, 2, 3), (1, 2), (2, 3), (1, 3) \}$

Cartesian Product of Two Sets

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Cartesian product of two sets $A$ and $B$, denoted $A \times B$ is the set of all ordered pairs whose first component is a member of $A$ and whose second component is a member of $B$. A table can be created by taking the Cartesian product of a set of rows and a set of columns.

If $A_{1}, A_{2},....., A_{m}$ are non empty sets, then the cartesian product of them is the set of all ordered m-tuples.

Example: 

If $A$ = $\{ 1, 2, 3 \}$ and $B$ = $\{ x, y, z \}$, find $A \times B$?

Solution: 

Given $A$ = $\{ 1, 2, 3 \}$

$B$ = $\{ x, y, z \}$

$A \times B$ = $\{ (1, x), (1, y), (1, z), (2, x), (2, y), (2, z), (3, x), (3, y), (3, z) \}$

Relations

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A relation $R$ from $A$ to $B$ is a subset of the cartesian product $A \times B$. If $A$ = $B$, then $R$ is said to be a relation on $A$.

The domain of $R$ is the set $\{ x | (x, y)$ is in $R$ for some $y$ in $B \}$. If the domain of $R$ is the set $A$, then $R$ is said to be total. Otherwise, $R$ is said to be partial.

The range of $R$ is the set $\{ y | (x, y)$ is in $R$ for some $x$ in $A \}$. The range of $R$ at $x$, denoted $R(x)$, is the set $\{ y | (x, y)$ is in $R \}$.