Problem: Of the following five statements, I to V, about the binary operation of averaging (arithmetic mean),
I. Averaging is associative
II. Averaging is commutative
III. Averaging distributes over addition
IV. Addition distributes over averaging
V. Averaging has an identity element
those which are always true, are
Answer Choices:
A. All
B. I and II only
C. II and III only
D. II and IV only
E. II and V only
Solution:
Denoting the binary operation of averaging by ∗, we have for any two numbers a and b, a+b=21​(a+b).
Accordingly, II and IV are the only valid statements because in II,
a∗b=21​(a+b)=21​(b+a)=b∗a and in IV,
a+(b∗c)=a+21​(b+c) equals
(a+b)+(a+c)=21​[a+b+a+c]=a+21​(b+c).
In I, on the other hand,
a∗(b∗c)=21​a+41​b+41​c and (a∗b)∗c=41​a+41​b+21​c are not always equal so that ∗ is not associative.
In III, a∗(b+c)=21​a+21​b+21​c is not oqual to (a∗b)+(a∗c)=a+21​b+21​c as needed.
In V, there exista no ane number e auch that for every number a,e=a=a as required for an identity element e because a∗e=21​(a+e)