Answer to Question #140410 in Discrete Mathematics for James macharia

Question

Out of 300 students taking discrete mathematics, 60 take coffee, 27 take cocoa 36 take tea, 17 take tea only, 47 take chocolate only, 7 take chocolate and cocoa ,3 take chocolate, tea and cocoa ,20 take cocoa only ,2 take tea, coffee and chocolate, 30 take coffee only, 9 take tea and chocolate whereas 12 take tea and coffee . Express this information on a venn diagram.

Answer 1

We note from the diagram below no. of tea drinkers $=17+a+b+c+d+e+f=36.$ Hence $a+b+c+d+e+f= 36-17=19.$ Also cocoa and chocolate $=d+e+h+i=7.$

Hence $a+b+c+d+h\leq 19+7=26.$ Also cocoa drinkers $20+b+d+g+h+i+e=27\Rightarrow b+d+g+h+i+e=7.$ Hence

$b+g+(d+e+h+i)=7\Rightarrow b+g+7=7\Rightarrow b=g=0.$ Hence $a+b+c+d+h+g\leq 26+0=26.$ But no of coffee drinkers =60. Whereas no. of coffee drinkers is $30+a+b+c+d+g+h\leq 26+30=56\Rightarrow \Leftarrow .$