13albrad
05.12.2019 •
Mathematics
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
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Ответ:
(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:
Commutative property of addition Additive identity property of additionCommutative property of property of multiplication over additionI. [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])
Ответ:
god
Step-by-step explanation: