ISRO2011-27

Question

Which one of the following is true?

• A. $R \cap S = ( R \cup S) - [(R-S) \cup (S-R)]$
• B. $R \cup S = ( R \cap S) - [(R-S) \cup (S-R)]$
• C. $R \cap S = ( R \cup S) - [(R-S) \cap (S-R)]$
• D. $R \cap S = ( R \cup S) \cup (R-S)$

Answer 1

A.    $R \cap S = ( R \cup S) - [(R-S) \cup (S-R)]$

{ 2 } = { 1 , 2 , 3 } - [ {1} ∪ {3} ]

{ 2 } = { 2 }

B.   $R \cup S = ( R \cap S) - [(R-S) \cup (S-R)]$

{ 1 , 2 , 3}  ≠ { 2 } - [ 1,3 ]

C.    $R \cap S = ( R \cup S) - [(R-S) \cap (S-R)]$

{ 2 }  ≠ { 1 , 2 , 3 } - [ 1 ∩ 3 ]

D.    $R \cap S = ( R \cup S) \cup (R-S)$

{ 2 } =  { 1 , 2 , 3 } ∪{1}

{ 2 }  ≠ { 1 , 2 , 3 }

Hence.Option(A)  $R \cap S = ( R \cup S) - [(R-S) \cup (S-R)]$ is the correct choice.

Answer 2

Answer is option [a] 

Comments

alternatively we can verify by making venn diagrams

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yes..

Answer 3

option A is correct simply draw venn diagrams with two sets R and S having some part in common

R intersection S will be =R union S (which cover all area of R and S)-(symmetric difference R-S union symmetric diff S-R)

Answer 4

Set Theory Formula:

$A - B = A\bigcap B{}' = A.B'$

With this formula

Option a becomes:

$(R + S) - [RS' + SR'] =(R + S)[RS' + SR']'

=(R + S)(R' + S)(S' + R)

=[R + (S.S')](R' + S)

=[R + 0](R' + S)

=R(R' + S)

=RS =R \cap S =LHS$