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: \[ \boxed{(2), (4)} \]