Symmetric Difference Operator and tricks

if P and Q are set then Symmetric difference Δ\Delta is given by

P \Delta Q = (P \cup Q) - (P \cap Q)

for Simplicity :

\cap = .            // product operator

\cup = +       //  Sum operator

and 

A- B = A. B^{c} 

Now 

P \Delta Q = (P \cup Q) - (P \cap Q)

               = (P+ Q) -(PQ)

              = (P+Q).(PQ)^{c}

              = (P+Q).(P^{c} + Q^{c})

              = PQ^{^{c}} + P^{^{c}}Q

If you remember it is the formula for XOR 

The symmetric difference Δ\Delta is nothing but XOR operator.

for convenient we can apply all the property of XOR where Symmetric Difference Operator is used. 

 

Example:-

Let E , F and G be finite sets and  

X = (E \cap F) - (F \cap G)

Y = (E - (E \cap G)) - (E -F)

 

What is the relation between X and Y  ?

X = EF . (FG)^{c}

        = EF . (F^{c} + G^{c})

       = EFG^{c}

Similarly 

Y = EFG^{c} 

X=Y = EFG^{c}

0Comment
On-To function

2. OnTo Function:-

If each and every element of the range has at least one preimage in the domain.

What it means is: let us say there are 4 elements in Range and they all are mapped to at least one element in Domain.

So for this condition, there must be at least 4 elements in Domain. Why not less than 4?

Because we already know that each element of the domain can be mapped with at most one element of Range. If there are less than 4 elements then one element of Range is left unmapped. That's why we need at least 4 elements in Domain.

 

3. Bijection Function:-

A function is bijection iff

  • The function is one-one.
  • The function is onto.

If A and  B are finite set then Bijection from A to B is possible only iff 

|A| =|B| =n

Total no.of Bijection Possible is:  n!

 

4.Inverse function:-

let f: A\rightarrow B and  a function f^{_{-1}}: B\rightarrow A  is called the inverse of f.

 

Necessary Condition for Inverse function:-

  • The original function should be one-one.
  • The original function should be onto.
  • Inverse exist only for Bijective Function.

 

 

 

 

 

 

 

0Comment
One -One function

1.One - One function or Injection:-

  • If every element of the domain is mapped with unique elements of Range.
  • If the domain has 'm' elements then range must have 'n' elements such that                         n\geq m

let |A| = n and |B| = m then the total number of one -one function from A to B is:- 

       $^m P_n$.

 

example:  Is f(x) = x^{2} is One -One function?

Solution:-

If anywhere you are asked about one-one function what you need to do is like this:

f(x) = x^{2}

f(y)=y^{2}

now equate them:

f(x) = x^{2} =f(y)=y^{2}

or 

x^{2} =y^{^{2}}

x^{2} - y^{2} =0

(x+y)(x-y) =0

or

x=y or x=-y Since we are getting two value for x ,so it is not One -One function.

 

 

 

0Comment
Function Introduction

The special type of relation is called as a function.

A relation from A to B is called as a function A to B represented as

f: A\rightarrow B   only if, for each  a \varepsilon A , we can assign a unique  element 'b' such that b \varepsilon B

where a is called Domain element and b is called as range elements.

Some characteristics of function:

1. All the elements of the domain must have a unique image in the range.

2. It is not compulsory that all images in Range have a preimage in the domain.

3. A relation is failed to become a function if any element in domain doesn't have an image in the range.

4. A relation is failed to become a function if any element in the domain has more than one image in the range.

 

 

 Total Number of function:

let say |A| = m and |B| = n then the number of function from A to B is  n^{^{^{m}}} 

 

 

 

 

 

 

0Comment
Set Theory Basics(1)

Subset: B is called a subset of A if the number of elements in B is less than or equal to A. Ex:

A ={1,2,3,4}

B={2,3,4}

We can say that B ⊆ A

 

Proper Subset:

B is called the proper subset of A if A contains at least one element that B doesn't contain.  So A is also called as a proper superset. B⊂ A.

 

 The cardinality of the set: Number of elements present in a set is called the cardinality of the set. Ex:

X={a,b,c,d,e,f,g}

Cardinality is denoted of X is written as |X|

|X| = 7

 

Null Set: A set having no element is called a null set. It is denoted by {} or Ø

But {Ø} is not a null set.

 

Must read Topics and Examples:-

0Comment
Basics of set theory | Set | Power Set | Cartesian Product

Set - A collection of unordered and distinct objects.
Examples:
N = {0, 1, 2, 3, .....} , the set of natural numbers
Z = {....., -2, -1, 0, 1, 2,....}, the set of integers
Z+ = {1, 2, 3,......}, the set of positive integers
Q = { p/q | p ε Z , and q≠0}, the set of rational numbers
Q+ = { x ε R | x=p/q, for some positive integers p and q }, the set of all positive rational numbers
R, the set of all real numbers

Power Set: 
Given a set S, the power set of S is the set of all subsets of the set S. It is denoted by P(S). 
Examples:
S = {0 ,1 ,2}
P(S) = P({0, 1, 2}) = {Φ, {0}, {1}, {2}, {0 ,1}, {0, 2}, {1, 2}, {1, 2, 3}}.
Note:
S = Φ
P(S) = {Φ}
P(P(S)) = {Φ, {Φ}}
Cartesian Product: The cartesian product of sets A and B, denoted by A x B, is the set of all ordered pairs (a, b), where aε A and b ε B. Hence,
A x B = {(a, b) | a ε A ∧ b ε B}
Example:
A = {1, 2}
B = {a, b, c}
A x B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)}

Set Identities:

0Comment