GATE CS 2024 Set 1 — Question 52

Let us analyze each statement:Statement (A): Associativity of $\diamond$
Check if $(a \diamond b) \diamond c = a \diamond (b \diamond c)$.
LHS: $(a \diamond b) \diamond c = (a + 2b) \diamond c = (a + 2b) + 2c = a + 2b + 2c$
RHS: $a \diamond (b \diamond c) = a \diamond (b + 2c) = a + 2(b + 2c) = a + 2b + 4c$
Since $a + 2b + 2c \neq a + 2b + 4c$ generally, $\diamond$ is NOT associative.Statement (B): Associativity of $\Box$
$a \Box b = ab$ (standard multiplication).
Multiplication of integers is associative: $(ab)c = a(bc)$.
Thus, $\Box$ is associative.Statement (C): Distributivity of $\diamond$ over $\Box$
Check if $a \diamond (b \Box c) = (a \diamond b) \Box (a \diamond c)$.
LHS: $a \diamond (bc) = a + 2bc$
RHS: $(a + 2b) \Box (a + 2c) = (a + 2b)(a + 2c) = a^2 + 2ac + 2ab + 4bc$
LHS $\neq$ RHS generally. Thus, $\diamond$ does NOT distribute over $\Box$.Statement (D): Distributivity of $\Box$ over $\diamond$
Check if $a \Box (b \diamond c) = (a \Box b) \diamond (a \Box c)$.
LHS: $a \Box (b + 2c) = a(b + 2c) = ab + 2ac$
RHS: $(ab) \diamond (ac) = ab + 2(ac) = ab + 2ac$
LHS = RHS. Thus, $\Box$ distributes over $\diamond$.Therefore, statements (B) and (D) are TRUE.