GATE CS 2024 Set 1 | Question 52

MSQ · 2 marks

Consider the operators \(\lozenge\) and \(\square\) defined by \(a \mathbin{\lozenge} b = a + 2b\), \(a \mathbin{\square} b = ab\), for positive integers. Which of the following statements is/are TRUE?

  1. Option A: Operator \(\lozenge\) obeys the associative law
  2. Option B: Operator \(\square\) obeys the associative law (correct)
  3. Option C: Operator \(\lozenge\) over the operator \(\square\) obeys the distributive law
  4. Option D: Operator \(\square\) over the operator \(\lozenge\) obeys the distributive law (correct)

Answer

(B), (D)

Explanation

  • (A) False: \((a \mathbin{\lozenge} b) \mathbin{\lozenge} c = a + 2b + 2c\), but \(a \mathbin{\lozenge} (b \mathbin{\lozenge} c) = a + 2(b + 2c) = a + 2b + 4c\).
  • (B) True: \(\square\) is ordinary multiplication, which is associative.
  • (C) False: \(a \mathbin{\lozenge} (b \mathbin{\square} c) = a + 2bc\), while \((a \mathbin{\lozenge} b) \mathbin{\square} (a \mathbin{\lozenge} c) = (a + 2b)(a + 2c)\). For \(a = b = c = 1\) these are 3 and 9.
  • (D) True: \(a \mathbin{\square} (b \mathbin{\lozenge} c) = a(b + 2c) = ab + 2ac = (a \mathbin{\square} b) \mathbin{\lozenge} (a \mathbin{\square} c)\). It also holds from the right, since \((b + 2c)a = ba + 2ca\).

So multiplication distributes over \(\lozenge\), because \(\lozenge\) is a linear combination of its operands, but not the other way round.

Answer: B, D