Question

If A, B and C are any three sets then prove the following using venn-diagram. A∩(BUC) = (A∩B) U (A∩C).

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

TO PROVE

Using Venn Diagram

$\sf{ \: A \cap \: (B \cup \: C) = ( A \cap B ) \cup \: (A \cap C)}$

PROOF

STEP : 1

First we draw vertical lines in the whole areas of B and C so as to represent ( B U C )

( Refer to the Figure 1 )

STEP : 2

Now we draw horizontal lines in the whole area A . Thus the double hatched area in Figure 2 represents area common to

A and ( B U C ) that is A ∩ ( B U C )

STEP : 3

Again we draw horizontal lines in the area common to A and B so as to represent

( A ∩ B ) ( Refer to the Figure 3 )

STEP : 4

Now we draw vertical lines in the area common to A and C so as to represent

( A ∩ C )

Then the whole hatched area in Figure 4 represents ( A ∩ B ) U ( A ∩ C )

STEP : 5

Hence we observe that the double hatched area in Figure 2 is equal to the total hatched area in Figure 4

Hence proved

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LEARN MORE FROM BRAINLY

If A and B are two non empty sets then A∪B=B∩A iff 

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