Determine whether the function is one-to-one or onto?
Determine whether the following function is one-to-one or onto, justifying
with proof or counterexample
g : Z → Z given by g(n) = 2n + 2, if n even or n, if n odd
- PopeLv 71 month ago
Let p and q be integers such that g(p) = g(q).
Suppose p and q both even.
2p + 2 = 2q + 2
2p = 2q
p = q
Suppose p and q both odd.
p = q
Suppose p even and q odd.
2p + 2 = q
q = 2(p + 1)
That would make q an even number, which contradicts the hypothesis.
Therefore g(p) = g(q) only if p = q.
Function g is one-to-one.
Zero is an integer, but there exists no integer r such that g(r) = 0.
Function g is not onto.
- 1 month ago
A function is not a one-to-one function if at least two points of the domain are ... Proof: Suppose x1 and x2 are real numbers such that f(x1) = f(x2). ... On the other hand, to prove a function that is not one-to-one, a counter example has to be given. ... g(n) = 2n - 1 for all n Z. Prove that g is not onto by giving a counter example.
- Ian HLv 71 month ago
This link explains the concepts clearly with diagrams and examples.