Question

If A,B and C are three sets then prove that ---- A×(BuC)=(A×B)u(A×C)

Answer 1

Let,

$(x,\ y)\in [A\times (B\cup C)]\quad\longrightarrow\quad (1)$

This means,

$x\in A\land y\in (B\cup C)$

That is,

$x\in A\land (y\in B\lor y\in C)$

By distributive law,

$(x\in A\land y\in B)\lor (x\in A\land y\in C)$

Or,

$[(x,\ y)\in (A\times B)]\lor [(x,\ y)\in (A\times C)]$

Or,

$(x,\ y)\in [(A\times B)\cup(A\times C)]\quad\longrightarrow\quad (2)$

From (1) and (2) we just get,

$A\times (B\cup C)\subseteq(A\times B)\cup(A\times C)\quad\longrightarrow\quad (i)$

Now, let,

$(x,\ y)\in [(A\times B)\cup(A\times C)]\quad\longrightarrow\quad (3)$

Then,

$[(x,\ y)\in (A\times B)]\lor [(x,\ y)\in (A\times C)]\\\\(x\in A\land y\in B)\lor (x\in A\land y\in C)\\\\x\in A\land (y\in B\lor y\in C)\\\\x\in A\land y\in (B\cup C)$

That is,

$(x,\ y)\in [A\times (B\cup C)]\quad\longrightarrow\quad (4)$

From (3) and (4), we just get,

$A\times (B\cup C)\supseteq(A\times B)\cup(A\times C)\quad\longrightarrow\quad (ii)$

But from (i) and (ii) we actually get that,

$A\times (B\cup C)=(A\times B)\cup(A\times C)$

Hence Proved!

Answer 2

Answer:

A×(BUC) = (A*B) U (A*C)

Step-by-step explanation:

• A German mathematician by name Georg Cantor discovered the concept of the set theory.
• A set is defined as “when the set is having a number of objects there is whole of a definite distinct of the objects. They are a collection of well-defined objects.

• There are different standard forms of sets. They are:
• Set of natural numbers
• Set of whole numbers
• Set of integers
• Set of odd numbers
• Set of even numbers
• Set of real numbers

There are different kinds of sets. They are:

Finite set:

It is a set used fro counting the finite numbers of elements.

Example: set of number of students in a class.

Infinite set:

It is a set used for counting the infinite number of elements.

Example: counting of natural numbers.

Empty set: It is a set which has no element

Singleton set: It is a set with single elements.

There are other properties related to union of sets .They are:

1) commutative law:

it is defined by

A ∪ B = B ∪ A

2) Associative law:

it is defined by

A ∪ (B ∪ C) = (A ∪ B) ∪ C

3) Identity of law:

it is defined by,

A∪ϕ=AA∪ϕ=A

4) Idempotent law:

It is defined by

A∪A=A

Given that:

There are A, B and C are three sets.

To prove:

A×(BUC)=(A×B)U(A×C)

By applying the distributive law, we get:

A x (B U C) = A * [(B U C) –(B ∩ C)]

            = A * ( B UC) –A  (B ∩ C)

           = (A *B ) U (A*C) –(A * B) ∩ (A*C)

   Cancelling the common values, we get,

(A*B) U (A*C)

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