Sets, Relations and Functions

Sets

A set is well defined collection of distinct objects. the sets are generally denoted by capital alphabets A, B, C,........ and the elements of a set are denoted by small alphabets a,b,c....
N: Set of all natural numbers
R: Set of all real numbers
Z: Set of all integers
Q: Set of all rational numbers
C: Set of all complex numbers
Z+: Set of all positive integers

If an object 'a' belongs to a set 'A', then we write a $\in$ A and if it does not belong to 'A', we write it as a $\notin$ A

Methods for describing a set:-

1) Roster Method (Tabular Method)
Here a set is described by listing elements, seperated by commas, within brackets{ }
eg:- The set of vowels of English alphabets may be described as {a, e, i, o, u}

2) Rule Method (Set-builder method or property method)
Here the set is described by {x: p(x)} or {x/p(x)} where p(x) is the property which the element x must satisfy. The symbol '/' or ':' is read as 'such that'.
eg:- A = {x / x is a vowel in English alphabet}
Cardinal number or order of a set: The number of elements in the set and it is denoted by n(A) or |A|.
Singleton Set: A set consisting of a single element
Null set or Empty set or Void set: A set having no elements and is denoted by { } or $φ$
Subsets: If every element of a set A is also an element of another set B, then A is said to be a subset of B and is denoted by A $\subset$ B. In this case B is said to be super set of A and denoted by B $\supset$ A.

Note:
(1) Every set is a subset of itself.
(2) Null set is a subset of every set.
(3) The total number of subsets of a set containing n elements is $2^{n}$ .

Power Set: The collection of all possible subsets of a set. eg:- If A= {1,2} then P(A)= { $φ$ , {1}, {2], {1,2}}
Universal Set: The superset of all the sets in a given context and denoted by the letter 'U' or 'X'.
Similar or Equivalent sets: Two sets A and B are said to be equivalent if n(A) = n(B).
Union or Join of sets: The union of any two sets A and A, denoted by A $⋃$ B, is the set of all elements which belong either to A or to B or to both A and B.

If A= {1,2,3}, B= {2,3,4,5}, then A $⋃$ B = {1,2,3,4,5}
x $\in$ A $⋃$ B $\Leftrightarrow$ x $\in$ A or x $\in$ B
x $\in$ A $⋃$ B $\Leftrightarrow$ x $\notin$ A or x $\notin$ B

Intersection of Sets: The Intersection of any two sets A and B, denoted by A $⋃$ B, is defined as the set of all elements which are common to both A and B.
If A= {1,2,3}, B= {2,3,4,5} then A $⋃$ B = {2,3}
x $\in$ A $⋂$ B $\Leftrightarrow$ x $\in$ A and x $\in$ B

Disjoint Sets: Two sets A and B are said to be disjoint if A $⋂$ B = $φ$
Difference of Sets: The difference of any two sets A and B, denoted by A - B, is the set of elements of A which do not belong to B.
If A= {1,2,3,4} B= {3,4,5}, then A-B= {1,2}

Complement of a set: The complements of any set A, denoted by A' or $A^{c}$ , is the set of all elements in U which are not in A.

Symmetric difference: The symmetric difference of any two sets A and B, denoted by A $Δ$ B, is the set (A-B) $⋃$ (B-A)

Theorems
1. (i) U' = $φ$ , $φ$ ' = U
(ii) ${(A^{'})}^{'}$
= A
(iii) A $⋃$ A'= $⋃$
(iv) A $⋂$ A'= $φ$

2. (i) A $⋃$ B
(ii) A $⋂$ A= A
(iii) A $⋃$ $φ$ = A
(iv) A $\cap$ U= A

3. (i) A $\cup$ B = B $\cup$ A
(ii) A $\cap$ B = B $\cap$ A

4. (i) A $\cup$ (B $\cup$ C) = (A $\cup$ B) $\cup$ C
(ii) A $\cap$ (B $\cap$ C) = (A $\cap$ B) $\cap$ C

5. (i) A $\cup$ (B $\cap$ C) = (A $\cup$ B) $\cap$ (A $\cup$ C)
(ii) A $\cap$ (B $\cup$ C) = (A $\cap$ B) $\cup$ (A $\cap$ C)

6. (i) (A $\cup$ B)' = A' $\cap$ B'
(ii) (A $\cap$ B)' = A' $\cup$ B'

7. (i) A - (B $\cup$ C) = (A-B) $\cap$ (A-C)
(ii) A- (B $\cap$ C) = (A-B) $\cup$ (A-C)
(iii) A $\cap$ (B-C) = (A $\cap$ B) - (A $\cap$ C)
(iv) A $\cap$ (B $Δ$ C) = (A $\cap$ B) $Δ$ (A $\cap$ C)

8. (i) A- B = A $\cap$ B', B-A = B $\cap$ A', A-B= A-(A $\cap$ B)
(ii) A- B = A $\Leftrightarrow$ A $\cap$ B = $φ$
(iii) (A-B) $\cup$ B= A $\cup$ B
(iv) (A-B) $\cap$ B= $φ$
(v) A $\subset$ B $\Leftrightarrow$ B' $\subset$ A'
(vi) (A-B) $\cup$ ( B-A) = (A $\cup$ B) - (A $\cap$ B)

Results on number of elements in sets
(i) n (A - B) = n (A) - n (A $\cap$ B) or n (A-B) + n(A $\cap$ B) = n(A)
(ii) n(A $\cup$ B) = n(A) + n (B) - n(A $\cap$ B) = n(A) + n(B) if A and B are disjoint set
(iii) n(A $\cup$ B $\cup$ C) = n(A) + n(B) + n(C) - n(A $\cap$ B) - n (B $\cap$ C) - n (A $\cap$ C) + n(A $\cap$ B $\cap$ C)
(iv) n(A' $\cap$ B' $\cap$ C')= n(U) - n(A $\cup$ B $\cup$ C)
(v) n(A' $\cap$ B') = n(U) - n(A $\cup$ B)
(vi) n(A' $\cup$ B') = n(U) - n(A $\cap$ B)
(vii) n(A $Δ$ B) = n(A) + n(B) -2. n(A $\cap$ B)
(viii) No. of elements in exactly two of the seats A, B, C is equal to n(A $\cap$ B) + n(B $\cap$ C) + n(C $\cap$ A) - 3 . n(A $\cap$ B $\cap$ C)
(ix) No. of elements in exactly one of the sets A, B, C is equal to n(A) + n(B) + n(C) - 2 . n(A $\cap$ B) - 2 . n(B $\cap$ C) - 2 . n(C $\cap$ A) + 3. n(A $\cap$ B $\cap$ C)

Cartesian Product of two sets
The product of any two sets A and B, denoted by A x B, is defined as the set of all ordered pairs (x,y) such that x $\in$ A, y $\in$ B. If A= {a,b}, B= {c,d} then A x B = {(a,c), (a,b), (b,c), (b,d) }

Some results on cartesian product
(i) A x (B $\cup$ C) = (A x B) $\cup$ (A x C)
(ii) A x (B $\cap$ C) = (A x B) $\cap$ (A x C)
(iii) A x (B- C) = (A x B) - (A x C)
(iv) A x B = B x A $\Leftrightarrow$ A = B
(v) (A x B) $\cap$ (C x D) = (A $\cap$ C) x (B $\cap$ D)
(vi) A x (B' $\cup$ C')' = (A x B) $\cap$ (A x C)
(vii) Ax (B' $\cap$ C')' = (A x B) $\cup$ (A x C)
(viii) n(A x B) = n(A) . n(B)
(ix) If A and B have n elements in common, then A x B and B x A have $n^{2}$ elements in common.

Relation
A relation R from a set A to another set B is a subset of A x B. If (a,b) $\in$ R, we write ${}_{a}{R_{b}}$ and read as 'a is related to b'. If n(A) = m, n(B)= n, then n(AxB)= mn.

$∴$ Total number of relations from A to B is $2^{mn}$ .

Domain and Range of a relation
If R is relation from set A to another set B, then the domain of R is the set of all first co-ordinates of the members of R and the range of R is the set of all second co-ordinates of the members of R.Thus Dom(R) = {a: (a, b) $\in$ R} and Range (R) = {b: (a,b) $\in$ R}

Inverse relation
Let R be a relation from Set 'A' to another set 'B'. Then the inverse relation of R, denoted by $R^{- 1}$ , is a relation from B to A. Thus $R^{- 1}$ ={ (b,a) / (a,b) $\in$ R}

Relation on a set: A relation from a set 'A' to itself is called a relation on set A. ie. a subset of A x A.
Note:-
1) Since $φ$ $\subset$ A x A, it is a relation, is a void relation.
2) Since A x A $\subset$ A x A, it is a relation called universal relation.
Identity relation: A relation $|_{A}$ on A is called the identity relation, if every element of A is related to itself only. eg:- A= {1,2,3} then $|_{A}$ = {(1,1), (2,2), (3,3)}

Reflective relation: A relation R on a set A is said to reflective if (a,a) $\in$ R for all a $\in$ A.
Note:
1) The identity relation on a non-empty set A is always reflective. But a reflective relation is not necessarily identity relation on A.
2) The universal relation on a non-empty set A is reflective.

Symmetric relation: A relation R on a set A is said to be a symmetric relation if (a,b) $\in$ R $\Rightarrow$ (b,a) $\in$ R ie. aRb $\Rightarrow$ bRa for all a,b, $\in$ A.
Note:
1) The identity and universal relations on a non-empty set are symmetric relations.
2) A relation r on a set A is symmetric if R = $R^{- 1}$

Transitive relation: A relation R on a set A is said to be transitive if (a,b) $\in$
R and (b,c) $\in$ R $\Rightarrow$ (a,c) $\in$ R ie. aRb and bRc $\Rightarrow$ aRc for all a,b,c $\in$ A.
Note:-
The identity and universal relations on a non-empty set are transitive.

Equivalence Relation: A relation R on a set A is said to be an equivalence relation if it is reflexive, symmetric and transitive.
Note:-
1) If R and S are two equivalence relations on a set A, then R $\cap$ S is also an equivalence relation on A.
2) R $\cup$ S is not necessarily an equivalence relation on the set A.
3) If R is an equivalence relation on A then $R^{- 1}$ is also an equivalence relation.

Function (Mapping)

A function 'f' form a set A to another set B associates each elements of set A to a unique element of set B.
Note:-
1) A function from A to B is a subset of AxB.
2) Every function is a relation but every relation is not a function.

Domain, Co-domain and Range
Let f: A $\to$ B. Then the set A is known as the domain of f and the set B is known as the co-domain of f. The set of all images of elements of A is known as range of f. The image of any element 'x' is denoted as f(x).

One-one function(Injection)
A function f: A $\to$ B is said to be one-one if different elements of A have different images in B ie. if $X_{1}$ $\neq$ $X_{2}$ $\Rightarrow$ f( $x_{1}$ ) $\neq$ f( $x_{2}$ ).

Many-one function
A function f: A $\to$ B is said to be amny-one, if different elements of A have same images in B.

Onto function (surjection)
A function f: A $\to$ B is said to be onto, if every elements of B is an image of an element in A ie. Range and co-domain of 'f' are the same.

Into function
A function f: A $\to$ B is an into function if there is at least one element in B which is not an image of any element in A. That is range is a proper subset of co-domain.

Bijection
One - one and onto functions are called bijections.

Note:- If n(A) = m, n(B) = n then
(1) Total number of functions from A to B = $n^{m}$
(2) The number of onto functions from A to B

= $\sum_{r = 1}^{n}$ ${(- 1)}^{n - r}$ nCr. $r^{m}$ . where 1 $\leq$ n $\leq$ m

(3) The number of bijections from A to B= n! (m=n)= 0, otherwise
(4) The number of one one function from A into B is = nPm, n $\geq$ m = 0, otherwise
(5) The number of onto functions that can be defined from A to B is $2^{m}$ - 2, if n=2.

Composite of functions
Let f: A $\to$ B and g: B $\to$ C be two functions then a function gof: A $\to$ C defined by (gof) (x) = g [f(x)]. for all x $\in$ A is called the composition of f and g.

composition

Inverse of a function
If f: A $\to$ B is a bijection, then a function from B to A which associates each element y $\in$ B to its pre-image $f^{- 1}$ (y) $\in$ A. Such function is called inverse function of 'f' denoted by $f^{- 1}$

finverse

Signum function

f(x) = ${\frac{| X |}{\underset{0}{X}}$

x $\neq$ 0
x = 0 and is written as sgnx. Its domain is R and range is R{-1, 0,1}

Greatest integer function: The greatest integer function is denoted by [x] and its valve is the greatest integer less than or equal to x.

Even and odd function: A function f(x) is said to be even if f(-x) = f(x) and is odd if f(-x) = -f(x).

Unitary operation: A function F: A $\to$ A, where A is a set, is considered as a unitary operation, if an element of A is associated to each singelton subset {a} of A.

Binary Operation: Let S be a non-void set, A function f: S x S $\to$ S is considered as a binary operation and is denoted by '*'.

Properties
Let * be a binary operation on a set S
1. It is commutative if a*b = b*a, for all a, b, $\in$ S 2.
2. It is associative if (a*b)*c = a*(b*c), for all a, b, c $\in$ S
3. The element e $\in$ S is said to be an identity element if a*e = a = e*a, for all a $\in$ S.
4. If a* a' = e = a'* a for all a, a' $\in$ S. Then 'a' and 'a" are inverses of each other w.r.t. '*'.
5. Let * and 'o' be any two binary operations on a set S then * is said to be

(a) Left distributive over 'o' if a * (b o c) = (a*b) o (a*c)

(b) Right distributive over 'o' if (b o c)* a = (b*a) o (c*a)
for all a, b, c $\in$ S.

Results
Let S be a finite set containing 'n' elements then
(1) Total no. of binary operations on S is $n^{(n^{2})}$
(2) Total no. of commutative binary operations on S is $n^{\frac{n (n + 1)}{2}}$
(3) The total no. of non commutative binary operation on S is $n^{{(n}^{2})}$ - $n^{\frac{n (n + 1)}{2}}$ .