Menu Close

if-x-A-100-and-x-B-75-x-AUB-50-then-find-x-A-B-and-x-B-A-




Question Number 7324 by rohit meena last updated on 23/Aug/16
if x(A)=100  and x(B)=75  x(AUB)=50  then find x(A−B) and x(B−A)
$${if}\:{x}\left({A}\right)=\mathrm{100}\:\:{and}\:{x}\left({B}\right)=\mathrm{75} \\ $$$${x}\left({AUB}\right)=\mathrm{50} \\ $$$${then}\:{find}\:{x}\left({A}−{B}\right)\:{and}\:{x}\left({B}−{A}\right) \\ $$
Commented by Rasheed Soomro last updated on 23/Aug/16
Impossible situation  Let x(A∩B)=y  x(A−B)=100−y ,  x(B−A)=75−y  x(A−B)+x(A∩B)+x(B−A)=x(AUB)  (100−y)+y+(75−y)=50  175−y=50  y=175−50=125   But     y=x(A∩B)≤75  Hence impossible situation.
$${Impossible}\:{situation} \\ $$$${Let}\:{x}\left({A}\cap{B}\right)={y} \\ $$$${x}\left({A}−{B}\right)=\mathrm{100}−{y}\:,\:\:{x}\left({B}−{A}\right)=\mathrm{75}−{y} \\ $$$${x}\left({A}−{B}\right)+{x}\left({A}\cap{B}\right)+{x}\left({B}−{A}\right)={x}\left({AUB}\right) \\ $$$$\left(\mathrm{100}−{y}\right)+{y}+\left(\mathrm{75}−{y}\right)=\mathrm{50} \\ $$$$\mathrm{175}−{y}=\mathrm{50} \\ $$$${y}=\mathrm{175}−\mathrm{50}=\mathrm{125} \\ $$$$\:{But}\:\:\:\:\:{y}={x}\left({A}\cap{B}\right)\leqslant\mathrm{75} \\ $$$${Hence}\:{impossible}\:{situation}. \\ $$
Commented by rohit meena last updated on 23/Aug/16
kese
$${kese} \\ $$
Commented by Rasheed Soomro last updated on 24/Aug/16
A∪B contains all the members of A  so it must have at least 100 members(=x(A) )  It cannot have only 50 members.    x(A∪B)≥max( x(A),x(B) )  Also,  x(A∩B)≤min( x(A),x(B) )
$${A}\cup{B}\:{contains}\:{all}\:{the}\:{members}\:{of}\:{A} \\ $$$${so}\:{it}\:{must}\:{have}\:{at}\:{least}\:\mathrm{100}\:{members}\left(={x}\left({A}\right)\:\right) \\ $$$${It}\:{cannot}\:{have}\:{only}\:\mathrm{50}\:{members}. \\ $$$$ \\ $$$${x}\left({A}\cup{B}\right)\geqslant{max}\left(\:{x}\left({A}\right),{x}\left({B}\right)\:\right) \\ $$$${Also}, \\ $$$${x}\left({A}\cap{B}\right)\leqslant{min}\left(\:{x}\left({A}\right),{x}\left({B}\right)\:\right) \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *