Logical Venn Diagrams Questions and Answers

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.

Answer: B

The answer is E, the region inside the square and the circle but outside the triangle.

Players are the square and graduates are the circle, so players who are graduates lie in the overlap of square and circle. That overlap holds two regions, D and E. Region D also lies inside the triangle, which stands for girls, so it is excluded by 'but not girls'. Region E lies outside the triangle and is the one left.

A is a girl only, B is a girl who is a player but not a graduate, and D is a girl who is a player and a graduate.

Answer: D

The answer is the diagram with a circle inside a larger circle and a separate third shape, option (d).

An encyclopaedia is a kind of book, so its set lies entirely inside the set of books. A magazine is neither a book nor an encyclopaedia, so its set stands apart from both.

Option (a) and option (c) make the magazine overlap the books or the encyclopaedias, which is wrong because no magazine is a book. Option (b) shows all three as unconnected, which misses that every encyclopaedia is a book.

Answer: A

The answer is diagram 1, three circles each overlapping the other two.

A person can be a tennis fan, a cricket player and a student all at once, and equally any two of these without the third. Every pair of classes therefore shares members, and the three classes share some members too, which is the three-circle diagram where all circles overlap.

Diagram 2 puts tennis fans and cricket players wholly inside students, which would make every fan and every player a student. Diagram 3 leaves the first and third circles apart, so no tennis fan could be a cricket player or the reverse. Diagram 4 places one class entirely inside another, which is not true of any pair.