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David Walnut
MATH 290002
December 14
th
, 2010
Writing Assignment 10
1. Let A and B be finite sets with A~B.
Suppose f: A
B
a. If f is onetoone, show that f is onto B.
Let A and B be two finite Sets and A
≈
B.
Assume f is a function from A to B.
Assume f is one to one, we will show that f is onto.
Assume f is not not onto, we will show f is not onetoone.
Since A is finite, A
≈
N
k
and A contains k elements.
Since B is finite B
≈
N
k
and since f is not onto b, there exists
element b such that b is not in the range of A.
Since A
≈
N
k
and B
≈
N
k,
then B{b}
≈
N
k.
Define
Define
b. If f is onto B, prove that f is onetoone.
Let A and B be two finite Sets.
Assume f is a function from A to B.
Assume function f is onto B, we will show f is onetoone.
Since f is onto, for each b
∈
B, there exists a
∈
A such that
f(a) = b.
Therefore each element b of B has a preimage in A.
Since we assumed A
≈
B, then the cardinality of A equals
the cardinality of B,
Hence, each element of B has only one preimage in A.
Therefore, different elements of A have different images in
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 Fall '08
 Alligood,K
 Advanced Math, Sets

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