by Brian Bosse, Copyright February 25, 2008, all rights reserved. 45 views
There are three types of mapping functions: injections, surjections and bijections.
INJECTION - An injection is a function taking each element of a set A to a unique different element in set B, but perhaps not to all elements of set B. In short, if f is an injection, then whenever f(x)=f(y), then x=y. An injection is sometime called a "one-to-one" mapping.
Let A={1, 2, 3} and B={1, 2, 3, 4, 5, 6} Let f: A → B, where f(x)=2x. The 1, 2 and 3 in A will map to the 2, 4, and 6 in B; however, elements 1, 3 and 5 in B are not mapped to. Therefore, f is an injection.
SURJECTION - A surjection is a function taking each element of a set A to one element in set B, but some of the elements of set A may share the same element in set B. In short, if f is an surjection, then for any b ∈ B, there exists an a ∈ A such that b=f(a). A surjection is sometime referred to as being "onto".
Let A={1, 2, 3, 4, 5, 6} and B={1, 2, 3} Let f: A → B, where if x is even then f(x)=1, if x<5 and is odd, then f(x)=2, otherwise f(x)=3. The 2, 4 and 6 in A will map to the 1 in B. The 1 and 3 in A will map to the 2 in B, and the 5 in A will map to the 3 in B. Because multiple elements in A map to the same element in B, f is a surjection.
BIJECTION - A bijection is a function that is both an injection (one-to-one) and a surjection (onto).
Let A={1, 2, 3, 4, 5, 6} and B={1, 2, 3, 4, 5, 6} Let f: A → B, where f(x)=7-x. The 1 in A maps to the 6 in B. The 2 in A maps to the 5 in B, etc… f is onto and one-to-one, and as such is a bijection.
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