A special-interest group has conducted a survey concerning a ban on hand guns. Note: A rifle is a gun, but it is not a hand gun. The survey yielded the following results for the 1000 households that responded.267 own a hand gun.437 own a rifle.493 supported the ban on hand guns.136 own both a hand gun and a rifle.206 own a rifle but no hand gun and do not support the ban on hand guns.70 own a hand gun and support the ban on hand guns.48 own both a hand gun and a rifle and also support the ban on hand guns.How many of the surveyed households:(a) only own a hand gun and do not support the ban on hand guns?(b) do not own a gun and support the ban on hand guns?(c) do not own a gun and do not support the ban on hand guns?

Answer:a)109 of the surveyed households only own a hand gun and do not support the ban on hand guns.b)328 of the surveyed households do not own a gun and support the ban on hand gunsc )112 of the surveyed households do not own a gun and do not support the ban on handguns.Step-by-step explanation:To solve this problem, we must build the Venn's Diagram of this set.I am going to say that:-The set A represents those who own a hand gun.-The set B represents those who own a rifle-The set C represents those who support the ban of handguns.-D represents those who do not own a gun and do not support the ban on hand guns.We have that:[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]In which [tex]a[/tex] are those that only own a hand gun, [tex]A \cap B[/tex] are those who own both a handgun and a rifle, [tex]A \cap C[/tex] are those who own a hand gun and support the ban on handguns, and [tex]A \cap B \cap C[/tex] are those who own both a hand gun and a rifle, and also support the ban on hand guns.By the same logic, we have:[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]In which [tex]b[/tex] are those who only own a rifle, and [tex]B \cap C[/tex] are those who own a rifle and support the ban on handguns.[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]In which [tex]c[/tex] are those who support the ban on hand guns and do not own any handguns.This diagram has the following subsets:[tex]a,b,c,D(A \cap B), (A \cap C), (B \cap C), (A \cap B \cap C)[/tex]There were 1000 households surveyed, so:[tex]a + b + c + D + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 1000[/tex]We start finding the values from the intersection of three sets.Solution:48 own both a hand gun and a rifle and also support the ban on hand guns.[tex]A \cap B \cap C = 48[/tex]70 own a hand gun and support the ban on hand guns:[tex](A \cap C) + (A \cap B \cap C) = 70[/tex][tex]A \cap C = 70 - 48[/tex][tex]A \cap C = 22[/tex]206 own a rifle but no hand gun and do not support the ban on hand guns.[tex]b = 206[/tex].136 own both a hand gun and a rifle:[tex](A \cap B) + (A \cap B \cap C) = 136[/tex][tex]A \cap B = 136 - 48[/tex][tex]A \cap B = 88[/tex]493 supported the ban on hand guns.[tex]C = 493[/tex]437 own a rifle:[tex]B = 437[/tex]267 own a hand gun:[tex]A = 267[/tex]a) How many of the surveyed households only own a hand gun and do not support the ban on hand guns?This is the value of [tex]a[/tex], that we can find from the following equation:[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex][tex]267 = a + 88 + 22 + 48[/tex][tex]a = 267 - 158[/tex][tex]a = 109[/tex]109 of the surveyed households only own a hand gun and do not support the ban on hand guns.b) How many of the surveyed households do not own a gun and support the ban on hand guns?This is the value of [tex]c[/tex], that we can find from the following equation:[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex][tex]493 = c + 22 + (B \cap C) + 48[/tex]Here, we also have to find [tex]B \cap C[/tex], that are those who own a rifle and support the ban on hand guns. We can find this from the following equation:[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex][tex]437 = 206 + (B \cap C) + 88 + 48[/tex][tex]B \cap C = 437 - 342[/tex][tex]B \cap C = 95[/tex]So[tex]493 = c + 22 + (B \cap C) + 48[/tex][tex]493 = c + 22 + 95 + 48[/tex][tex]c = 493 - 165[/tex][tex]c = 328[/tex]328 of the surveyed households do not own a gun and support the ban on hand guns.a) How many of the surveyed households  do not own a gun and do not support the ban on hand guns?This is the value of d, that can be found from the following equation:[tex]a + b + c + D + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 1000[/tex][tex]109 + 206 + 328 + D + 88 + 22 + 95 + 40 = 1000[/tex][tex]D = 1000 - 888[/tex][tex]D = 112[/tex]112 of the surveyed households do not own a gun and do not support the ban on handguns.