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Difficulty: EasyVenn Diagrams and Set-Based Data

In a survey of 100100 residents in a locality, 6060 residents read Newspaper X, 5050 read Newspaper Y, and 2020 read both Newspaper X and Newspaper Y. How many residents read neither Newspaper X nor Newspaper Y?

  1. 1010Answer
  2. B
    2020
  3. C
    3030
  4. D
    4040

Answer

The number of residents who read neither Newspaper X nor Newspaper Y is 10.
The total number of residents who read at least one newspaper is given by N(X∪Y)=N(X)+N(Y)−N(X∩Y)=60+50−20=90N(X \cup Y) = N(X) + N(Y) - N(X \cap Y) = 60 + 50 - 20 = 90. Thus, the number of residents who read neither newspaper is 100−90=10100 - 90 = 10.

Step-by-Step Solution

1
Calculate the number of residents who read at least one newspaper using the Inclusion-Exclusion principle.
∣X∪Y∣=60+50−20=90|X \cup Y| = 60 + 50 - 20 = 90
Simply adding total readers of X and Y double-counts the residents who read both newspapers.
2
Subtract the number of residents reading at least one newspaper from the total surveyed population.
Neither = 100−90=10100 - 90 = 10
The universe of residents consists of those reading at least one newspaper and those reading neither.

Key Concept

Principle of Inclusion-Exclusion for two sets
Estimated Time:1m 0s
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Topics

Venn Diagrams and Set-Based Data

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