Answer to Question #327119 in Discrete Mathematics for joshua

Question

In a group of 35 ex-athletes, 17 play golf, 20 go cycling, and 12 do yoga. Exactly 8

play golf and go cycling, 8 play golf and do yoga, 7 go cycling and do yoga, and 4 do

all three activities. How many of the ex-athletes do none of these activities?

Answer 1

$A-golf\\B-cycling\\C-yoga\\\left| \varOmega \right|=35\\\left| A \right|=17,\left| B \right|=20,\left| C \right|=12\\\left| A\cap B \right|=8,\left| A\cap C \right|=8,\left| B\cap C \right|=7\\\left| A\cap B\cap C \right|=4\\\left| \bar{A}\cap \bar{B}\cap \bar{C} \right|=\left| \varOmega \right|-\left| A\cup B\cup C \right|=\\=\left| \varOmega \right|-\left( \left| A \right|+\left| B \right|+\left| C \right|-\left| A\cap B \right|-\left| A\cap C \right|-\left| B\cap C \right|+\left| A\cap B\cap C \right| \right) =5\\=35-\left( 17+20+12-8-8-7+4 \right) =5$