GATE CS 2024 Set 1 Q42 — Monoids and Groups

We analyze each option for positive integers under the operations $a \diamond b = a + 2b$ and $a \square b = ab$:

• Option A: For associativity of $\diamond$, we compare $(a \diamond b) \diamond c$ and $a \diamond (b \diamond c)$: $$(a \diamond b) \diamond c = (a + 2b) + 2c = a + 2b + 2c$$ $$a \diamond (b \diamond c) = a + 2(b + 2c) = a + 2b + 4c$$ Since $a + 2b + 2c \neq a + 2b + 4c$ for positive integers, $\diamond$ is not associative.
• Option B: For associativity of $\square$, we have: $$(a \square b) \square c = (ab)c = abc$$ $$a \square (b \square c) = a(bc) = abc$$ Since integer multiplication is associative, $\square$ is associative.
• Option C: For distributivity of $\diamond$ over $\square$, we check if $a \diamond (b \square c) = (a \diamond b) \square (a \diamond c)$: $$\text{LHS} = a \diamond (bc) = a + 2bc$$ $$\text{RHS} = (a + 2b) \square (a + 2c) = (a + 2b)(a + 2c)$$ Since LHS $\neq$ RHS, this distributive law does not hold.
• Option D: For distributivity of $\square$ over $\diamond$, we check if $a \square (b \diamond c) = (a \square b) \diamond (a \square c)$: $$\text{LHS} = a \square (b + 2c) = a(b + 2c) = ab + 2ac$$ $$\text{RHS} = (ab) \diamond (ac) = ab + 2ac$$ Since LHS $=$ RHS, this distributive law holds.