GATE 2024 CS1 Q.52: solution and answer

GATE 2024 · CS1Q.52 · 2 marks · MSQ

Consider the operators ◊ and □ defined by $a ◊ b = a + 2 b$ , $a □ b = ab$ , for positive integers. Which of the following statements is/are TRUE?

(A)
Operator ◊ obeys the associative law
(B)
Operator □ obeys the associative law
(C)
Operator ◊ over the operator □ obeys the distributive law
(D)
Operator □ over the operator ◊ obeys the distributive law

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Answer and solution

ANSWER

Statements (B) and (D) are true.

Official answer: B, DSource: Official answer key, GATE 2024 CS1 (IISc Bengaluru)

Write $a ◊ b = a + 2 b$ and $a □ b = ab$ on positive integers. Test each law with general symbols, and use a counterexample to refute a law.
(A) Associativity of $◊$ : $(a ◊ b) ◊ c = (a + 2 b) + 2 c = a + 2 b + 2 c$ , while $a ◊ (b ◊ c) = a + 2 (b + 2 c) = a + 2 b + 4 c$ . These differ because $c \neq = 0$ . For example $(1 ◊ 1) ◊ 1 = 3 ◊ 1 = 5$ but $1 ◊ (1 ◊ 1) = 1 ◊ 3 = 7$ . So (A) is false.
(B) Associativity of $□$ : $(a □ b) □ c = (ab) c = a (b c) = a □ (b □ c)$ , because multiplication of integers is associative. So (B) is true.
(C) $◊$ over $□$ requires $a ◊ (b □ c) = (a ◊ b) □ (a ◊ c)$ . The left side is $a + 2 b c$ ; the right side is $(a + 2 b) (a + 2 c) = a^{2} + 2 ab + 2 a c + 4 b c$ . With $a = b = c = 1$ the left side is $3$ and the right side is $9$ . So (C) is false.
(D) $□$ over $◊$ requires $a □ (b ◊ c) = (a □ b) ◊ (a □ c)$ . The left side is $a (b + 2 c) = ab + 2 a c$ ; the right side is $ab + 2 (a c) = ab + 2 a c$ . They are equal for all $a, b, c$ . Since $□$ is commutative, the right-distributive form $(b ◊ c) □ a = (b □ a) ◊ (c □ a)$ holds too. So (D) is true.
The true statements are (B) and (D).

KEY IDEA

To test a law, expand both sides with general symbols and compare. One counterexample is enough to refute a law. Distributivity of one operator over another needs the left side to expand to exactly the right side.

WATCH OUT FOR

Calling $◊$ associative because $a + 2 b$ looks like addition, when the factor 2 on the second argument breaks it. Another slip is expanding $(a + 2 b) (a + 2 c)$ wrongly in (C) and finding a match.

From the GATE 2024 CS1 paper. Every question of this paper · Download the full paper · All 7 Groups questions, with notes

GATE 2024 CS1 Q.52: solution and answer

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