Logical Venn Diagrams Questions and Answers

Answer: C

140 candidates appeared.

The permitted sets of three are A B C, A B D, A B F and C D E: A and B go together and A excludes E, and without A and B the only set that keeps F away from C and D is C D E. E appears only in C D E, so 70 candidates took C D E. F appears only in A B F, so 35 took A B F. C appears in A B C and C D E, so A B C has 90 − 70 = 20. D appears in A B D and C D E, so A B D has 85 − 70 = 15. Check: A = 20 + 15 + 35 = 70.

Each candidate offered exactly three subjects, and the subject counts add up to 70 + 70 + 90 + 85 + 70 + 35 = 420. So the number of candidates is 420 / 3 = 140. The same total comes from 20 + 15 + 35 + 70 = 140. The figure 380 is the sum of the first five counts, which is not the number of candidates.

Answer: B

20 candidates offered A + B + C.

The permitted sets of three are A B C, A B D, A B F and C D E: A and B go together and A excludes E, and without A and B the only set that keeps F away from C and D is C D E. E appears only in C D E, so 70 candidates took C D E. F appears only in A B F, so 35 took A B F. C appears in A B C and C D E, so A B C has 90 − 70 = 20. D appears in A B D and C D E, so A B D has 85 − 70 = 15. Check: A = 20 + 15 + 35 = 70.

So the combination A B C was taken by 20 candidates. The figure 15 belongs to A B D, 35 to A B F, and 70 to C D E.

Answer: D

70 candidates combined C with D.

The permitted sets of three are A B C, A B D, A B F and C D E: A and B go together and A excludes E, and without A and B the only set that keeps F away from C and D is C D E. E appears only in C D E, so 70 candidates took C D E. F appears only in A B F, so 35 took A B F. C appears in A B C and C D E, so A B C has 90 − 70 = 20. D appears in A B D and C D E, so A B D has 85 − 70 = 15. Check: A = 20 + 15 + 35 = 70.

C and D occur together only in the set C D E, and 70 candidates took it. Nobody can take C and D with A and B in the same set, because three subjects only are allowed.

Answer: C

35 candidates offered B with F.

The permitted sets of three are A B C, A B D, A B F and C D E: A and B go together and A excludes E, and without A and B the only set that keeps F away from C and D is C D E. E appears only in C D E, so 70 candidates took C D E. F appears only in A B F, so 35 took A B F. C appears in A B C and C D E, so A B C has 90 − 70 = 20. D appears in A B D and C D E, so A B D has 85 − 70 = 15. Check: A = 20 + 15 + 35 = 70.

B and F occur together only in A B F, and F is taken by 35 candidates in all, all of them in A B F, so 35 offered B with F.

Answer: C

The figure is diagram 3, three circles each overlapping the other two.

Musicians and Dancers are both groups of people, and the groups overlap, because a person can be a musician and a dancer at once. Neither group lies wholly inside Women, because there are men who are musicians and men who are dancers. Women overlap each of the two groups, since some women are musicians, some are dancers and some are both. Three circles that each cut across the other two show exactly this.

Diagram 1 puts Musicians and Dancers wholly inside Women and apart from each other, so no one can be both and no man can belong. Diagram 2 nests the three, making every musician a dancer and every dancer a woman. Diagram 4 keeps the two overlapping but still puts them wholly inside Women, which would make every musician and dancer a woman.

Answer: D

The number is 16.

Read each region by the shapes it lies in. Players are the circle, Lawyers the triangle and Singers the rectangle. Singers who are players but not lawyers lie inside both the rectangle and the circle but outside the triangle. That region holds 16.

The other numbers belong to other regions: 24 lies in the circle and triangle but not the rectangle (players and lawyers who are not singers); 17 lies in the circle only (players who are neither singers nor lawyers); 12 lies in the triangle and rectangle but outside the circle (singers and lawyers who are not players). The 11 in the circle, triangle and rectangle is the players who are both singers and lawyers, so it is left out.

Answer: A

The consistent option is (A).

The model is the subtractive colour scheme: Yellow and Magenta overlap in Red, Yellow and Cyan in Green, Magenta and Cyan in Blue, and all three in Black.

In each option the rectangle is Y, the blob is M, and both the small leaf and the tilted square are Cy. Option (A) labels every overlap correctly: leaf and rectangle G, leaf and blob Bl, leaf, rectangle and blob K, rectangle and blob R, rectangle and square G, and rectangle, blob and square K.

(B) labels the rectangle and blob overlap Bl instead of R. (C) puts Bl where leaf, rectangle and blob meet, which must be K. (D) labels the three-way overlap with the square R and the rectangle and square overlap Bl, which should be K and G.

Answer: B

The number with both diabetes and high BP is 15.

Everyone with high cholesterol also has high BP, and no one has both cholesterol and diabetes. So the 45 high-BP patients split into three groups:

• high BP only: 20
• high BP with high cholesterol (all 10 cholesterol patients): 10
• high BP with diabetes: x

\[ 20 + 10 + x = 45 \Rightarrow x = 15 \]

The total of 75 then leaves 30 patients with diabetes only, which is consistent. 20 is the high-BP-only count, 10 the cholesterol count, and 0 would leave the BP total at 30.

Answer: A

A takes B and the reverse, A excludes E, and C or D excludes F. The allowed sets of three are:

• with A and B: A B C, A B D, A B F (A B E is barred because A excludes E);
• without A and B, three from C, D, E and F: only C D E, since every other set puts F with C or D.

E appears only in C D E, so all 70 E candidates took C D E. F appears only in A B F, so 35 took A B F. C appears in A B C and C D E, so A B C has 90 − 70 = 20. D appears in A B D and C D E, so A B D has 85 − 70 = 15. Check: A = 20 + 15 + 35 = 70.

So 4 combinations were permitted: ABC, ABD, ABF and CDE. A count of 5 or more comes from allowing A B E, or a set of C, D, E and F that pairs F with C or D, both barred.

Answer: A

The number is 2.

An Indian leader who is not a singer lies inside both the Indian and the Leader circles but outside the Singer circle. That region is numbered 2.

4 is inside Indian and Singer (an Indian singer who is not a leader), 5 is inside Leader and Singer but not Indian, and 7 is a singer only. Region 3 would be an Indian leader who is also a singer, which the question excludes.