Question

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

• A. Operator $\diamondsuit$ obeys the associative law
• B. Operator $\square$ obeys the associative law
• C. Operator $\diamondsuit$ over the operator $\square$ obeys the distributive law
• D. Operator $\square$ over the operator $\diamondsuit$ obeys the distributive law

Answer 1

The Correct Option is B

Step 1: Associativity of $\diamondsuit$.
To check if $\diamondsuit$ is associative: $$(a \diamondsuit b) \diamondsuit c = (a + 2b) + 2c = a + 2b + 2c,$$ $$a \diamondsuit (b \diamondsuit c) = a + 2(b + 2c) = a + 2b + 4c.$$ Since the results are not equal, $\diamondsuit$ is not associative. Step 2: Associativity of $\square$.
To check if $\square$ is associative: $$(a \square b) \square c = (ab) \square c = (ab)c = abc,$$ $$a \square (b \square c) = a \square (bc) = a(bc) = abc.$$ Since the results are equal, $\square$ is associative. Hence, (2) is true. Step 3: Distributivity of $\square$ over $\diamondsuit$.
To check if $\square$ is distributive over $\diamondsuit$: $$a \square (b \diamondsuit c) = a \cdot (b + 2c) = ab + 2ac,$$ $$(a \square b) \diamondsuit (a \square c) = (ab) \diamondsuit (ac) = ab + 2ac.$$ Since the results are equal, $\square$ is distributive over $\diamondsuit$. Hence, (3) is false. Step 4: Distributivity of $\diamondsuit$ over $\square$.
To check if $\diamondsuit$ is distributive over $\square$: $$a \diamondsuit (b \square c) = a + 2(bc),$$ $$(a \diamondsuit b) \square (a \diamondsuit c) = (a + 2b)(a + 2c) \neq a + 2(bc).$$ Since the results are not equal, $\diamondsuit$ is not distributive over $\square$. Hence, (4) is true. Final Answer: $$\boxed{(2), (4)}$$