GATE CS 2024 Set 1 Q.52: Monoids and Groups

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

Operator $\diamond$ obeys the associative law

Operator $\square$ obeys the associative law

Operator $\diamond$ over the operator $\square$ obeys the distributive law

Operator $\square$ over the operator $\diamond$ obeys the distributive law

Official answer and solution

Official answer: B and D
From: Official GATE 2024 answer key

Solution

Test associativity and distributivity directly using the algebraic definitions $a \diamond b = a + 2b$ and $a \square b = ab$ .

(A) is FALSE: $(a \diamond b) \diamond c = (a + 2b) + 2c = a + 2b + 2c$ , whereas $a \diamond (b \diamond c) = a + 2(b + 2c) = a + 2b + 4c \neq a + 2b + 2c$ for positive integers.
(B) is TRUE: Standard integer multiplication $a \square b = ab$ is associative: $(ab)c = a(bc)$ .
(C) is FALSE: $a \diamond (b \square c) = a + 2bc$ , whereas $(a \diamond b) \square (a \diamond c) = (a + 2b)(a + 2c) = a^2 + 2ab + 2ac + 4bc \neq a + 2bc$ .
(D) is TRUE: Checking both left and right distributivity of $\square$ over $\diamond$ : $a \square (b \diamond c) = a(b + 2c) = ab + 2ac = (ab) \diamond (ac) = (a \square b) \diamond (a \square c)$ $(b \diamond c) \square a = (b + 2c)a = ba + 2ca = (b \square a) \diamond (c \square a)$

Final answer

Options (B), (D)

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2024 Computer Science and Information Technology - Set 1 (CS1)
Question: 52 of 65
Section: Computer Science and Information Technology

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