Question

If the operation is defined for all integers a and b by a@b = a + b - ab, which of the following statements must be true for all integers a, b and c?

I. a@b = b@a
II. a@0 = a
III. (a@b)@c = a@(b@c)

(A) I only
(B) II only
(C) I and II only
(D) I and III only
(E) I, II and III

Answer 1

(E) I, II and III

Explanation:

Given,

a@b = a + b - ab ∀ a, b ∈ Z ( set of all integers ),

For solving this question we need to remember the following properties of integers:

• Commutative property of addition
• Additive identity property of addition
• Commutative property of multiplication
• Distributive property of multiplication over addition

I. a@b = a + b - ab,

b@a = b + a - ba = a + b - ab

Thus, a@b = b@a

II. a@0 = a + 0 - a × 0

= a + 0 - 0

= a

III. (a@b)@c = (a@b)+c - (a@b)c

= (a + b - ab) + c - (a + b - ab)c

= a + b - ab + c - ac - bc + abc

= a + b + c - ab - bc - ac + abc,

Now, a@(b@c) = a@( b + c - bc )

= a + (b + c - bc) - a(b+c - bc)

= a + b + c - bc - ab - ac + abc

= a + b + c - ab - bc - ac + abc.

Thus, (a@b)@c = a@(b@c)