For the set N of natural numbers and a binary operation f : N x N → N, an…
2006
For the set N of natural numbers and a binary operation f : N x N → N, an element z ∊ N is called an identity for f, if f (a, z) = a = f(z, a), for all a ∊ N. Which of the following binary operations have an identity?
f (x, y) = x + y - 3
f (x, y) = max(x, y)
f (x, y) = xy
- A.
I and II only
- B.
II and III only
- C.
I and III only
- D.
None of these
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Correct answer: A
Answer: f(x,y)=x+y-3 and f(x,y)=max(x,y) have identities; the exponentiation operation does not.
For f(x,y)=x+y-3: Solve a+z-3=a for z. This gives z=3. Also 3+a-3=a, so 3 is a two-sided identity.
For f(x,y)=max(x,y): An identity must satisfy max(a,z)=a for all a, so z must be less than or equal to every natural number. Thus z is the least element of the natural numbers (for the usual convention this is 1, or 0 if that convention is used). That least element is a two-sided identity.
For f(x,y)=x^y (exponentiation): If a two-sided identity z existed, we would need a^z = a for every a, which forces z=1 for general a. But then z^a = 1^a = 1, which is not equal to a for a>1. Hence no element z satisfies both a^z=a and z^a=a for all a, so exponentiation has no two-sided identity.
Therefore the operations with identities are the first and the second only.