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?

Let the two binary operations on positive integers be:

Let us evaluate each statement:

• Option (A): Associativity of $\Diamond$ requires $(a \Diamond b) \Diamond c = a \Diamond (b \Diamond c)$.

  $$(a \Diamond b) \Diamond c = (a + 2b) \Diamond c = (a + 2b) + 2c = a + 2b + 2c$$

  $$a \Diamond (b \Diamond c) = a \Diamond (b + 2c) = a + 2(b + 2c) = a + 2b + 4c$$

  Since $a + 2b + 2c

e a + 2b + 4c$for positive integers,$\Diamond$ is not associative. Thus, (A) is FALSE.

• Option (B): Associativity of $\square$ requires $(a \square b) \square c = a \square (b \square c)$.

  $$(a \square b) \square c = (ab) \square c = (ab)c = abc$$

  $$a \square (b \square c) = a \square (bc) = a(bc) = abc$$

  Since multiplication is associative, $(a \square b) \square c = a \square (b \square c)$. Thus, (B) is TRUE.

• Option (C): Distributivity of $\Diamond$ over $\square$ requires $a \Diamond (b \square c) = (a \Diamond b) \square (a \Diamond c)$.

  $$\text{LHS} = a \Diamond (bc) = a + 2bc$$

  $$\text{RHS} = (a \Diamond b) \square (a \Diamond c) = (a + 2b)(a + 2c) = a^2 + 2ac + 2ab + 4bc$$

  Since $$\text{LHS}$

e $\text{RHS}$$ in general, (C) is FALSE.

• Option (D): Distributivity of $\square$ over $\Diamond$ requires $a \square (b \Diamond c) = (a \square b) \Diamond (a \square c)$.

  $$\text{LHS} = a \square (b + 2c) = a(b + 2c) = ab + 2ac$$

  $$\text{RHS} = (ab) \Diamond (ac) = ab + 2(ac) = ab + 2ac$$

  Since $\text{LHS} = \text{RHS}$, $\square$ distributes over $\Diamond$. Thus, (D) is TRUE.

Therefore, the correct statements are B and D.