5) Let S = { n belongs to natural numbers | n Ξ 1 (mod 3) }. Give an example of a bijective function f: S --> N.
Choose 3x + 1, where x is natural number /and the range belongs or a subset of natural numbers.
Now to prove the function is injective, we must show that if f(x1) = f(x2), then x1 = x2.
Suppose f(x1) = f(x2), then 3x1 + 1 = 3x2 + 1.
substract 1 from both sides.
3x1 = 3x2
divide both sides by 3.
x1 = x2
Therefore, the function is injective.
Let y belongs to the set of natural numbers, then we must show that there is an x belonging to S such that f(x) = y.
Choose x = (y - 1) / 3, then f(x) = f[(y - 1) / 3].
f(x) = 3 * [(y - 1) / 3] + 1
f(x) = y
Therefore f is sujective.
Since f is surjective and injective, f is bijective.
2007-03-11
14:38:26
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2 answers
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binqasimm20
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Mathematics