Functions The function f maps A to B f : A B f ( a ) = b where a A - PowerPoint PPT Presentation

Functions http://localhost/~senning/courses/ma229/slides/functions/slide01.html Functions http://localhost/~senning/courses/ma229/slides/functions/slide02.html Functions prev | slides | next prev | slides | next Let A and B be sets. A function f from A to B is an assignment of exactly one element of B to each element of A . We use several types of notation for functions: Functions The function f maps A to B f : A B f ( a ) = b where a A and b B 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 of 1 09/07/2003 04:37 PM 1 of 1 09/07/2003 04:37 PM Functions http://localhost/~senning/courses/ma229/slides/functions/slide03.html Functions http://localhost/~senning/courses/ma229/slides/functions/slide04.html Functions Functions prev | slides | next prev | slides | next Some examples of functions: If f is a function from A to B then A is the domain and B is the codomain of f . If f ( a )= b then b is the image of a and a is the preimage of b . 1. f ( x ) = x + 1, x Z . The range of f is the set of all f ( a ) for each a A . f ( x ) x 0 Bob 1 Bill A is the domain of f , B is the codomain, and the range of f is the 2 Belinda 2. set of all elements in B that elements in A are mapped to. 3. f ( a ) = 1, a R . 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 of 1 09/07/2003 04:37 PM 1 of 1 09/07/2003 04:37 PM

Functions http://localhost/~senning/courses/ma229/slides/functions/slide05.html Functions http://localhost/~senning/courses/ma229/slides/functions/slide06.html Functions Functions prev | slides | next prev | slides | next Given If f 1 and f 2 are functions from A to R then A ={ a , b , c , d }, B ={1,2,3,4} 1. ( f 1 + f 2 )( x ) = f 1 ( x ) + f 2 ( x ) 2. ( f 1 f 2 )( x ) = f 1 ( x ) f 2 ( x ) which of the following are functions? define the sum and product of f 1 and f 2 . These new functions are functions from A to R as well. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 of 1 09/07/2003 04:37 PM 1 of 1 09/07/2003 04:37 PM Functions http://localhost/~senning/courses/ma229/slides/functions/slide07.html Functions http://localhost/~senning/courses/ma229/slides/functions/slide08.html Functions Functions prev | slides | next prev | slides | next If f is a function from A to B and S A then f ( S ) is a subset of B and A function f is one-to-one or injective if and only if f ( x )= f ( y ) implies x = y for all x and y in the domain of f . is defined A function that is one-to-one is an injection . f ( S ) = { f ( s ) | s S }. Suppose A ={0,1,2,...,9} and f ( a ) = a +10. If B ={10,11,12,...,19} or any set containing this set then f is a function from A to B . If S ={2,4,6} then f ( S )={12,14,16} and f ( S ) B . one-to-one not one-to-one 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 of 1 09/07/2003 04:37 PM 1 of 1 09/07/2003 04:37 PM

� � Functions http://localhost/~senning/courses/ma229/slides/functions/slide09.html Functions http://localhost/~senning/courses/ma229/slides/functions/slide10.html Functions Functions prev | slides | next prev | slides | next Whether or not a function is one-to-one can depend on it’s domain. If every element of the codomain of a function is the image of some element in the domain then the function is said to be onto , or is For example, f ( x )=sin( x ) is not one-to-one if the domain is R but is called a surjection . one-to-one if the domain is - /2 x /2. A function f : A B is onto or surjective if and only if b B a A with f ( a )= b . Is f ( x ) = x 2 one-to-one? (answer) A function that is onto is a surjection . 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 onto not onto 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 of 1 09/07/2003 04:37 PM 1 of 1 09/07/2003 04:37 PM Functions http://localhost/~senning/courses/ma229/slides/functions/slide11.html Functions http://localhost/~senning/courses/ma229/slides/functions/slide12.html Functions Functions prev | slides | next prev | slides | next If a function is both one-to-one and onto it is called a one-to-one Consider a function f : A B that is a one-to-one correspondence: in correspondence or a bijection . this case it is both one-to-one and onto. Exercise: Construct a function on the integers that is onto but not one-to-one. (possible solution) Exercise: Construct a function on the integers that is one-to-one but not onto. (possible solution) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Notice that if we turn the arrows around we have a function from B to A . This function is also a one-to-one correspondence. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 of 1 09/07/2003 04:37 PM 1 of 1 09/07/2003 04:37 PM

Functions http://localhost/~senning/courses/ma229/slides/functions/slide13.html Functions http://localhost/~senning/courses/ma229/slides/functions/slide14.html Functions Functions prev | slides | next prev | slides | next Let f be a one-to-one correspondence from A to B . The inverse Let g be a function g : A B and let f : B C . The composition of the function of f is the function that assigns to each element b of B an functions f and g is denoted f g and is defined element a of A such that f ( a )= b . We denote the inverse function f -1 . ( f g )( a ) = f ( g ( a )) f -1 ( b ) = a when f ( a ) = b . The identity function on A is i A : A A where i A ( a )= a for each a A . This is a one-to-one correspondence and so is invertible. It is, in fact, its own inverse. The function f : Z Z such that f ( a )= a -4 for each a Z is a one-to-one correspondence with f -1 ( a )= a +4. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 of 1 09/07/2003 04:37 PM 1 of 1 09/07/2003 04:38 PM Functions http://localhost/~senning/courses/ma229/slides/functions/slide15.html Functions http://localhost/~senning/courses/ma229/slides/functions/slide16.html Functions Functions prev | slides | next prev | slides | next Let A = B = { x | x is a positive integer } and let Recall that the cartesian product of A and B is f ( x ) = 2 x , g ( x ) = x +5. A x B = { ( a , b ) | a A and b B } What is ( f g )( x )? (answer) and notice that What is ( g f )( x )? (answer) { ( a , b ) | a A and b = f ( a ) } A x B Conclusion if f is a function f : A B . In other words, ( a , f ( a )) A x B for all a A . 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 of 1 09/07/2003 04:38 PM 1 of 1 09/07/2003 04:38 PM

Functions http://localhost/~senning/courses/ma229/slides/functions/slide17.html Functions http://localhost/~senning/courses/ma229/slides/functions/slide18.html Functions Functions prev | slides | next prev | slides | next The floor function assigns to the real number x the largest integer Let f be a function f : A B . The graph of function f is the set of ordered pairs that is less than or equal to x . It is denoted with x . The ceiling function assigns to the real number x the smallest { ( a , b ) | a A and f ( a ) = b } integer that is greater than or equal to x . It is denoted with x . Example: Let f ( x ) = 2 x -1 be a function on Z . The graph of f is Note that both of these functions map the reals to the integers: x : R Z , x : R Z . Question: Are these functions one-to-one? onto? invertible? 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 of 1 09/07/2003 04:38 PM 1 of 1 09/07/2003 04:38 PM