Discrete Mathematics in Computer Science Partial and Total Functions Malte Helmert, Gabriele R¨oger University of Basel Important Building Blocks of Discrete Mathematics Important building blocks: sets relations functions In principle, functions are just a special kind of relations: 2 f : N0 ! N0 with f (x) = x 2 relation R over N0 with R = f(x; y) j x; y 2 N0 and y = x g. Important Building Blocks of Discrete Mathematics Important building blocks: sets relations functions In principle, functions are just a special kind of relations: 2 f : N0 ! N0 with f (x) = x 2 relation R over N0 with R = f(x; y) j x; y 2 N0 and y = x g. Functional Relations Definition A binary relation R over sets A and B is functional if for every a 2 A there is at most one b 2 B with (a; b) 2 R. a 1 a 1 b 2 b 2 B B A c 3 A c 3 d 4 d 4 e e functional not functional Functions { Examples 2 f : N0 ! N0 with f (x) = x + 1 abs : Z ! N0 with ( x if x ≥ 0 abs(x) = −x otherwise 2 2 distance : R × R ! R with p 2 2 distance((x1; y1); (x2; y2)) = (x2 − x1) + (y2 − y1) Functions { Examples 2 f : N0 ! N0 with f (x) = x + 1 abs : Z ! N0 with ( x if x ≥ 0 abs(x) = −x otherwise 2 2 distance : R × R ! R with p 2 2 distance((x1; y1); (x2; y2)) = (x2 − x1) + (y2 − y1) Functions { Examples 2 f : N0 ! N0 with f (x) = x + 1 abs : Z ! N0 with ( x if x ≥ 0 abs(x) = −x otherwise 2 2 distance : R × R ! R with p 2 2 distance((x1; y1); (x2; y2)) = (x2 − x1) + (y2 − y1) Partial Function { Example Partial function r : Z × Z 9 Q with ( n if d 6= 0 r(n; d) = d undefined otherwise Partial Functions Definition (Partial function) A partial function f from set A to set B (written f : A 9 B) is given by a functional relation G over A and B. Relation G is called the graph of f . We write f (x) = y for (x; y) 2 G and say y is the image of x under f . If there is no y 2 B with (x; y) 2 G, then f (x) is undefined. Partial function r : Z × Z 9 Q with ( n if d 6= 0 r(n; d) = d undefined otherwise n 2 has graph f((n; d); d ) j n 2 Z; d 2 Z n f0gg ⊆ Z × Q. Partial Functions Definition (Partial function) A partial function f from set A to set B (written f : A 9 B) is given by a functional relation G over A and B. Relation G is called the graph of f . We write f (x) = y for (x; y) 2 G and say y is the image of x under f . If there is no y 2 B with (x; y) 2 G, then f (x) is undefined. Partial function r : Z × Z 9 Q with ( n if d 6= 0 r(n; d) = d undefined otherwise n 2 has graph f((n; d); d ) j n 2 Z; d 2 Z n f0gg ⊆ Z × Q. Partial Functions Definition (Partial function) A partial function f from set A to set B (written f : A 9 B) is given by a functional relation G over A and B. Relation G is called the graph of f . We write f (x) = y for (x; y) 2 G and say y is the image of x under f . If there is no y 2 B with (x; y) 2 G, then f (x) is undefined. Partial function r : Z × Z 9 Q with ( n if d 6= 0 r(n; d) = d undefined otherwise n 2 has graph f((n; d); d ) j n 2 Z; d 2 Z n f0gg ⊆ Z × Q. Partial Functions Definition (Partial function) A partial function f from set A to set B (written f : A 9 B) is given by a functional relation G over A and B. Relation G is called the graph of f . We write f (x) = y for (x; y) 2 G and say y is the image of x under f . If there is no y 2 B with (x; y) 2 G, then f (x) is undefined. Partial function r : Z × Z 9 Q with ( n if d 6= 0 r(n; d) = d undefined otherwise n 2 has graph f((n; d); d ) j n 2 Z; d 2 Z n f0gg ⊆ Z × Q. The domain of definition of f is the set dom(f ) = fx 2 A j there is a y 2 B with f (x) = yg. The image (or range) of f is the set img(f ) = fy j there is an x 2 A with f (x) = yg. f : fa; b; c; d; eg f1; 2; 3; 4g a 1 9 f (a) = 4; f (b) = 2; f (c) = 1; f (e) = 4 b 2 B domain fa; b; c; d; eg A c 3 codomain f1; 2; 3; 4g d 4 domain of definition dom(f ) = fa; b; c; eg e image img(f ) = f1; 2; 4g Domain (of Definition), Codomain, Image Definition (domain of definition, codomain, image) Let f : A 9 B be a partial function. Set A is called the domain of f , set B is its codomain. The domain of definition of f is the set dom(f ) = fx 2 A j there is a y 2 B with f (x) = yg. The image (or range) of f is the set img(f ) = fy j there is an x 2 A with f (x) = yg. domain of definition dom(f ) = fa; b; c; eg image img(f ) = f1; 2; 4g Domain (of Definition), Codomain, Image Definition (domain of definition, codomain, image) Let f : A 9 B be a partial function. Set A is called the domain of f , set B is its codomain. f : fa; b; c; d; eg f1; 2; 3; 4g a 1 9 f (a) = 4; f (b) = 2; f (c) = 1; f (e) = 4 b 2 B domain fa; b; c; d; eg A c 3 codomain f1; 2; 3; 4g d 4 e The image (or range) of f is the set img(f ) = fy j there is an x 2 A with f (x) = yg. image img(f ) = f1; 2; 4g Domain (of Definition), Codomain, Image Definition (domain of definition, codomain, image) Let f : A 9 B be a partial function. Set A is called the domain of f , set B is its codomain. The domain of definition of f is the set dom(f ) = fx 2 A j there is a y 2 B with f (x) = yg. f : fa; b; c; d; eg f1; 2; 3; 4g a 1 9 f (a) = 4; f (b) = 2; f (c) = 1; f (e) = 4 b 2 B domain fa; b; c; d; eg A c 3 codomain f1; 2; 3; 4g d 4 domain of definition dom(f ) = fa; b; c; eg e Domain (of Definition), Codomain, Image Definition (domain of definition, codomain, image) Let f : A 9 B be a partial function. Set A is called the domain of f , set B is its codomain. The domain of definition of f is the set dom(f ) = fx 2 A j there is a y 2 B with f (x) = yg. The image (or range) of f is the set img(f ) = fy j there is an x 2 A with f (x) = yg. f : fa; b; c; d; eg f1; 2; 3; 4g a 1 9 f (a) = 4; f (b) = 2; f (c) = 1; f (e) = 4 b 2 B domain fa; b; c; d; eg A c 3 codomain f1; 2; 3; 4g d 4 domain of definition dom(f ) = fa; b; c; eg e image img(f ) = f1; 2; 4g a 1 b 2 B A c 3 d 4 e Preimage The preimage contains all elements of the domain that are mapped to given elements of the codomain. Definition (Preimage) Let f : A 9 B be a partial function and let Y ⊆ B. The preimage of Y under f is the set f −1[Y ] = fx 2 A j f (x) 2 Y g. f −1[f1g] = f −1[f3g] = f −1[f4g] = f −1[f1; 2g] = Preimage The preimage contains all elements of the domain that are mapped to given elements of the codomain. Definition (Preimage) Let f : A 9 B be a partial function and let Y ⊆ B. The preimage of Y under f is the set f −1[Y ] = fx 2 A j f (x) 2 Y g. −1 a 1 f [f1g] = b 2 f −1[f3g] = B A c 3 f −1[f4g] = d 4 −1 e f [f1; 2g] = Total Functions Definition (Total function) A (total) function f : A ! B from set A to set B is a partial function from A to B such that f (x) is defined for all x 2 A. ! no difference between the domain and the domain of definition a 1 b 2 B A c 3 d 4 e Total Functions Definition (Total function) A (total) function f : A ! B from set A to set B is a partial function from A to B such that f (x) is defined for all x 2 A. ! no difference between the domain and the domain of definition a 1 b 2 B A c 3 d 4 e Total Functions Definition (Total function) A (total) function f : A ! B from set A to set B is a partial function from A to B such that f (x) is defined for all x 2 A.