VENN DIAGRAM INDEX 1 2 3 4 5

VENN DIAGRAM

INDEX 1. 2. 3. 4. 5. 6. DEFINATION OF VENN DIAGRAM OF A U B. VENN DIAGRAM OF A – B (ONLY A). VENN DIAGRAM OF B - A (ONLY B). VENN DIAGRAM OF AC.

VENN DIAGRAM o Most of the relationship between sets can be represented by means of diagrams which are known as Venn diagrams are named after the English logician, John Venn (1834 -1883). These diagrams consist of rectangles and closed curves usually circles. The universal set is represented usually by a rectangle and its subsets by circles.

1. Venn Diagram of AUB. A B AUB

2. Venn Diagram of A B U A B

3. Venn Diagram of A-B (ONLY A). o The part filled with color is A (only A). B A (A-B)

4. Venn Diagram of B-A (Only B) B-A A B

5. Venn Diagram of A Complement (Ac) A’ A’ A Ac

RELATIONS, FUNCTIONS DIFFERENT TYPES OF FUNCTIONS, DIFFERENCE BETWEEN RELATIONS AND FUNCTION

1. 2. 3. 4. 5. RELATION. CONSTITUENTS OF RELATION. FUNCTION. CONSTITUENTS OF FUNCTION. DIFFERENCE BETWEEN RELATION AND FUNCTION

5. EXAMPLE FOR RELATION AND FUNCTION. 6. VISUAL REPRESENTATION OF RELATION. 7. VISUAL REPRESENTATION OF FUNCTION. 8. DIFFERENCE IN WORDS. 9. BIBLIOGRAPHY.

DEFINITION: A Relation R from a set A to a set B is a subset of the Cartesian product A×B obtained by describing a relationship between the first element x and the second element y of the ordered pairs in a A×B. The second element is also called the Image of the First element. It is visually represented by arrow diagram.

1. 2. 3. IMAGE: The Image of an element x under a relation R is given by y, where (x, y) Є R. DOMAIN: The Domain of R is the set of all first elements of the ordered pairs in a Relation R. RANGE: The Range of the Relation R is the set of all second elements of the ordered pair in a relation R.

o DEFINITION: A Function f from a set A to a set B is a specific type of relation for which every element x of set A has one and only one image y in set B. We write f: A B, where f (x) = y.

1. 2. 3. A is the Domain and B is the codomain of f. The range of the function is the set of Images. A Real function has the set of real numbers or one of its subset both as its domain and as its range.

We can differentiate relation and function by visual representation. Lets take same example for relation and function.

EXAMPLE: Consider the two sets P={a, b, c} and Q={Ali, Bhanu, Binoy, Chandra}. The Cartesian product of P and Q has 12 ordered pairs which can be listed as P×Q={(a, Ali), (a, Bhanu), (a, Binoy)……, (c, Ch andra)}. Lets denote P and Q for Relation and A and B for functions.

VISUAL RERESENTATION FOR RELATION: P Q. Ali . A. B. C . Bhanu. Binoy. Chandra

VISUAL RERESENTATION FOR FUNCTION: A B. Ali. A. B. C . Bhanu. Binoy. Chandra

o The Difference between Relation and Function is that In relation every element x from set A need not have a unique image y in set B and In function every element x from set A have one and only one image y in set B.

FUNCTIONS o DIFFERENT TYPES OF FUNCTIONS

Functions : A relation F from set ‘A’ to ‘B’ (F : A to B) is said to be a function if every element of A has one &only one image in set B.

Some different type of functions are: -

1) 2) 3) 4) 5) 6) 7) Identity function Constant function Polynomial function Rational function Modulus function Signum function Greatest integer function

2) Constant function: Define the function f : R to R by y= f(x) =c, x belongs to R where c is a constant & each x belongs to R. here domain of f is R and its range is {c}.

1) Identity function : Let R be the set of real numbers. Define the real valued function f: R to R by y = f(x) =x for each x belongs to R. such a function is called the identity function. Here the domain and range of F are R.

3) Polynomial function: A function f : R to R is said to be polynomial function if for each x in R, y=f(x)=a 0+a 1 x+a 2 x 2+…. . +anxn, where n is a non–negative integer & a 0, a 1, a 2, …. , an belongs to R.

4) Rational function: Relation function are function of the type f(x)/ g(x), where f(x) & g(x) are polynomial function of x defined in a domain, where g(x) not equals to 0

5) Modulus function: A function f : R to R defined by f(x) = |x| for each x belongs to R is called modulus function. For each non-negative value of x, f(x) is equals to x. But for negative value of x, the value of f(x) is negative value of x.

7) Greatest integer function : A function f : R to R defined by f(x) = [x], x belongs to R assumes the value of the integer, less than or equal to x. such a function is called the greatest integer function. **END**

o The function f: R-R defined by f (x) = 1, if x >0 = 0, if x=0, = -1, if x <0 1 -4 -3 -2 -1 -1 x=-3 x=3