s4_2 - 2 PROPERTIES OF FUNCTIONS 111 2 Properties of...

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2. PROPERTIES OF FUNCTIONS 111 2. Properties of Functions 2.1. Injections, Surjections, and Bijections. Definition 2.1.1 . Given f : A B 1. f is one-to-one (short hand is 1 - 1 ) or injective if preimages are unique. In this case, ( a 6 = b ) ( f ( a ) 6 = f ( b )) . 2. f is onto or surjective if every y B has a preimage. In this case, the range of f is equal to the codomain. 3. f is bijective if it is surjective and injective (one-to-one and onto). Discussion We begin by discussing three very important properties functions defined above. 1. A function is injective or one-to-one if the preimages of elements of the range are unique. In other words, if every element in the range is assigned to exactly one element in the domain. For example, if a function is defined from a subset of the real numbers to the real numbers and is given by a formula y = f ( x ), then the function is one-to-one if the equation f ( x ) = b has at most one solution for every number b . 2. A function is surjective or onto if the range is equal to the codomain. In other words, if every element in the codomain is assigned to at least one value in the domain. For example, if, as above, a function is defined from a subset of the real numbers to the real numbers and is given by a formula y = f ( x ), then the function is onto if the equation f ( x ) = b has at least one solution for every number b . 3. A function is a bijection if it is both injective and surjective. 2.2. Examples. Example 2.2.1 . Let A = { a,b,c,d } and B = { x,y,z } . The function f is defined by the relation pictured below. This function is neither injective nor surjective. a ± < < < < < < < < < < < < < < < < < < x b / y c / z d 8 q q q q q q q q q q q q q
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2. PROPERTIES OF FUNCTIONS 112 Example 2.2.2 . f : A B where A = { a,b,c,d } and B = { x,y,z } defined by the relation below is a surjection, but not an injection. a / x b / y c / z d 8 q q q q q q q q q q q q q Example 2.2.3 . f : A B where A = { a,b,c,d } and B = { v,w,x,y,z } defined by the relation below is an injection, but not a surjection. a / v b / w c N N N N N N N N N N N N N x d N N N N N N N N N N N N N y z Example 2.2.4 . f : A B where A = { a,b,c,d } and B = { v,w,x,y } defined by the relation below both a surjection and an injection, and therefore a bijection. Notice that for a function to be a bijection, the domain and codomain must have the same cardinality. a ± < < < < < < < < < < < < < < < < < < v b / w c @ ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± x d / y Discussion
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2. PROPERTIES OF FUNCTIONS 113 The examples illustrate functions that are injective, surjective, and bijective. Here are further examples. Example 2.2.5 . Let f : [0 , ) [0 , ) be defined by f ( x ) = x . This function is an injection and a surjection and so it is also a bijection. Example 2.2.6 . Suppose f ( x ) = x 2 . If the domain and codomain for this function is the set of real numbers, then this function would be neither a surjection nor an injection. It is not a surjection because the range is not equal to the codomain. For example, there is no number in the domain with image - 1 which is an element of the codomain. It is not an injection since more than one distinct element in the domain is mapped to the same element in the codomain. For example,
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This note was uploaded on 11/27/2011 for the course MCH 108 taught by Professor Penelopekirby during the Fall '08 term at FSU.

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s4_2 - 2 PROPERTIES OF FUNCTIONS 111 2 Properties of...

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