here you are considering * as an intersection but in my question i am declare * as a Cartesian product.so during the designing of venn diagram can we consider cartesian product as an intersection.is it right way to solve this.....?

Cartesian Product and Intersection are two entirely different thing and Cartesian Product can not be represented by Venn Diagram. Please see the following link mentioned by Ranita.

@pritam:Ya.But Cartesian product is commonly applied for sets(or an element can contain more than one values) .And here i assumed it as simple Algebra(a variable can contain a unique value at a time),so i think Cartesian product will be equivalent to the product(multiplication).So product(multiplication) can be represented using venn diagrams as intersection.

I can tell you the formal proof,but without venn diagram.

(x,y)is all elements that belongs to A×(B∪C)
(x,y)∈A×(B∪C)
Obviously, x belongs to A and y belongs to (B∪C)
x∈A and y∈(B∪C)
x∈A and (y∈B or y∈C)
(x∈A and y∈B) or ( x∈A and y∈C)
(x,y)∈A×B or(x,y)∈A×C (Since, A×B contains all element of type (A,B))
(x,y)∈(A×B)∪(A×C)
Since,(x,y) belongs to (A×B)∪(A×C),means atleast (A×B)∪(A×C) contains (x,y) which is all elements that belongs to A×(B∪C)
therefore, A×(B∪C)⊂(A×B)∪(A×C)

Similarily take (x,y): all elements that belongs to (A×B)∪(A×C) and derive in same way (x,y)∈A×(B∪C)
and conclude (A×B)∪(A×C) ⊂ A×(B∪C).

Hence,A×(B∪C)=(A×B)∪(A×C).(because sets are subsets of each other, then they are equal.)

PS:INTERSECTION(^) in set,which is AND(*) in Algebra.And an number represent a region.

here you are considering * as an intersection but in my question i am declare * as a Cartesian product.so during the designing of venn diagram can we consider cartesian product as an intersection.is it right way to solve this.....?

yes.In set A.B represents both A and B should present,which is nothing but intersection.

@mahesh,

Cartesian Product and Intersection are two entirely different thing and Cartesian Product can not be represented by Venn Diagram. Please see the following link mentioned by Ranita.

@pritam:Ya.But Cartesian product is commonly applied for sets(or an element can contain more than one values) .And here i assumed it as simple Algebra(a variable can contain a unique value at a time),so i

thinkCartesian product will be equivalent to the product(multiplication).So product(multiplication) can be represented using venn diagrams as intersection.X.(Y−Z) can be drawn as follows:

<dot> =

Similarly, (X.Y)−(X.Z) can be drawn like this:

<minus> =

Hence, we can say that X.(Y−Z) = (X.Y)−(X.Z)

you are taking * as an intersection but it is the Cartesian product of X and Y.so i thing this is not the right way to proof it.

I don't think Cartesian product can be represented using Venn diagrams. You can try the methods proposed in this link to prove the given expression without using examples.

http://math.stackexchange.com/questions/362139/how-to-prove-the-distribu...

I can tell you the formal proof,but without venn diagram.

(x,y)is all elements that belongs to A×(B∪C)

(x,y)∈A×(B∪C)

Obviously, x belongs to A and y belongs to (B∪C)

x∈A and y∈(B∪C)

x∈A and (y∈B or y∈C)

(x∈A and y∈B) or ( x∈A and y∈C)

(x,y)∈A×B or(x,y)∈A×C (Since, A×B contains all element of type (A,B))

(x,y)∈(A×B)∪(A×C)

Since,(x,y) belongs to (A×B)∪(A×C),means atleast (A×B)∪(A×C) contains (x,y) which is all elements that belongs to A×(B∪C)

therefore, A×(B∪C)⊂(A×B)∪(A×C)

Similarily take (x,y): all elements that belongs to (A×B)∪(A×C) and derive in same way (x,y)∈A×(B∪C)

and conclude (A×B)∪(A×C) ⊂ A×(B∪C).

Hence,A×(B∪C)=(A×B)∪(A×C).(because sets are subsets of each other, then they are equal.)