GATE CS 1998 Q2.4 — Sets, Relations and Functions

We can solve this problem using the Principle of Inclusion-Exclusion (PIE) for three sets.

Let:

• $E$ be the set of people who speak English $\implies |E| = 18$
• $H$ be the set of people who speak Hindi $\implies |H| = 15$
• $K$ be the set of people who speak Kannada $\implies |K| = 22$

We are given the intersection sizes for pairs of languages:

• People speaking both English and Hindi: $|E \cap H| = 9$
• People speaking both Hindi and Kannada: $|H \cap K| = 11$
• People speaking both Kannada and English: $|K \cap E| = 13$

Assuming that everyone in the room speaks at least one of the three languages, the total number of people is the union of the three sets:

$$|E \cup H \cup K| = 28$$

According to the Principle of Inclusion-Exclusion:

$$|E \cup H \cup K| = |E| + |H| + |K| - (|E \cap H| + |H \cap K| + |K \cap E|) + |E \cap H \cap K|$$

Substitute the given values into the formula:

$$28 = 18 + 15 + 22 - (9 + 11 + 13) + |E \cap H \cap K|$$

$$28 = 55 - 33 + |E \cap H \cap K|$$

$$28 = 22 + |E \cap H \cap K|$$

$$|E \cap H \cap K| = 28 - 22 = 6$$

Thus, the number of people who speak all three languages is 6.