GATE CSE 2024 | Set 1 | Question: 42

Question

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

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

Answer 1

@ is said to obey associative law iff a @ (b @ c) = (a @ b) @ c.

@ is said to obey distributive law over # iff a @ (b # c) = (a @ b) # (a @ c)

A. $a \diamond (b \diamond c) = a + 2b + 4c$

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

$\implies a \diamond (b \diamond c) \neq (a \diamond b) \diamond c$

B. $a \square (b \square c) = abc$

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

$\implies a \square (b \square c) = (a \square b) \square c$

C. $a \diamond (b \square c) = a + 2bc$

$(a \diamond b) \square (a \diamond c) = (a+2b)(a+2c)$

$\implies a \diamond (b \square c) \neq (a \diamond b) \square (a \diamond c)$

D. $a \square (b \diamond c) = a(b+2c) = ab + 2ac$

$(a \square b) \diamond (a \square c) = ab + 2ac$

$\implies a \square (b \diamond c) = (a \square b) \diamond (a \square c)$

Answer - B, D