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Official PYQ · 2015Set 3 · Q24Propositional and First Order LogicMCQ1 mark

GATE CS 2015 Set 3 Q24 — Propositional and First Order Logic

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In a room there are only two types of people, namely Type 1\text{Type 1} and Type 2\text{Type 2}. Type 1\text{Type 1} people always tell the truth and Type 2\text{Type 2} people always lie. You give a fair coin to a person in that room, without knowing which type he is from and tell him to toss it and hide the result from you till you ask for it. Upon asking the person replies the following

"The result of the toss is head if and only if I am telling the truth"

Which of the following options is correct?

Choose one option
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Answer: A

To solve this puzzle, let us define two propositions:

  • HH: The result of the coin toss is head.
  • TT: The person is telling the truth.

The statement made by the person is:
"The result of the toss is head if and only if I am telling the truth"

Mathematically, this statement can be represented as: S:H↔TS: H \leftrightarrow T

Let us analyze the two possible cases for the person's type:

  1. Case 1: The person is of Type 1 (always tells the truth)
    • Since the person is a truth-teller, the proposition TT is True.
    • Because they always tell the truth, their statement SS must also be True.
    • Thus, H↔TrueH \leftrightarrow \text{True} is True\text{True}, which implies HH must be True (the result is head).

  2. Case 2: The person is of Type 2 (always lies)
    • Since the person is a liar, the proposition TT is False.
    • Because they always lie, their statement SS must be False.
    • Thus, H↔FalseH \leftrightarrow \text{False} is False\text{False}, which implies HH must be True (the result is head).

In both possible cases, the result of the coin toss is guaranteed to be head.

Therefore, the correct option is A.

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