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

(E) I, II and III

Step-by-step explanation:

Given,

[email protected] = a + b - ab ∀ a, b ∈ Z ( set of all integers ),

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

I. [email protected] = a + b - ab,

[email protected] = b + a - ba = a + b - ab

Thus, [email protected] = [email protected]

II. [email protected] = a + 0 - a × 0

= a + 0 - 0  

= a

III. ([email protected])@c = ([email protected])+c - ([email protected])c

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

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

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

Now, [email protected]([email protected]) = [email protected]( 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, ([email protected])@c = [email protected]([email protected])

n=9

step-by-step explanation:

7 times a number = 7n

decreased by 13 = 7n-13

same as is =

5 times the number = 5n

increased by 5 = 5n+5

7n-13 = 5n+5

2n=18

n=9