CS1022 Computer Programming & Principles

1. CS1022Computer Programming & Principles Lecture 1 Functions

2. Plan of lecture • Why functions? • Inverse relations and composition of relations (background to functions) • Functions • Terminology • Digraphs and graphs of functions • Properties of functions CS1022

3. Why functions? • Functions capture transformations • Input arguments used to compute output results • Input/output is key to computing • Advantages: • No commitments to computers/architectures • Abstract, clean, compact, focussed • Functions represent computations • Underpinned by -calculus (a kind of “interpreter”) • Programming paradigm: functional languages • What is x's age? Given a value for x, output the result: What is x's age?(Bill) => 34 CS1022

4. Procedural vs. functional programs • Factorial: 5! = 5  4  3 2 1 = 120 • Formally: or CS1022

5. Inverse relations • Given a binary relation R between sets A and B • The inverse relation R–1 between A and B is R–1 (b, a) : (a, b)  R • For instance, the inverse of relation is a parent of on the set of People is the relation is a child of • Graphically, the inverse of a binary relation is obtained by reversing all arrows in the digraph of original relation R (p,1), (p,2), (q,2) R–1 (1, p), (2, p), (2, q) R R–1 1 1 2 2 p p q q CS1022

6. Composition of relations (1) • Let R be a binary relation between sets A and B • Let S be a binary relation between sets B and C • The composition of R and S, represented as SR, is a binary relation between A and C given by SR(a,c) : aA, cC, aR b, bS c for some bB • The new relation relates elements of A and C using elements of B as “intermediaries” CS1022

7. Composition of relations (2) • If R is a relation “is a sister of” on a set of people • If S is a relation “is the mother of” on the same set • The compositionSR is SR(a,c) : ais a sister of b, bis mother of c SR(a,c) : ais a sister of themother of c SR(a,c) : ais an aunt of c • What about SS? SS(a,c) : ais mother ofb, bis mother of c SS(a,c) : ais mother ofthe mother of c SS(a,c) : ais a grandmother of c CS1022

8. Composition of relations (3) • Let R and S be binary relations defined as Then aR 1 and 1 Sy implies (a,y)  SR aR 2 and 2 Sx implies (a,x)  SR aR 3 and 3 Sx implies (a,x)  SR bR 2 and 2 Sx implies (b,x)  SR R(a,1), (a,2), (a,3), (b,2) 2 1 2 3 S(1,y), (2,x), (3,x) R SR S 1 x x y y 3 a a b b CS1022

9. Composition of relations w/ matrix • Relations represented as a matrices for • It easy to see relationship properties by inspecting the matrix, e.g. relationship is reflexive where it is 1 on the middle diagonal. • Computers view relations as a two dimensional array • How to compute compositions using matrices? • Problem: • Given the logical matrices of two relations R and S • Compute the logical matrix of their composition SR • Result is logical (or Boolean) matrix product CS1022

10. Composition with matrix (2) • Let Aa1,a2,,an Bb1,b2,,bm Cc1,c2,,cp • Relations R relates A and B S relates B and C • The logical matrix M representing R is given by • M(i, j)  T (true) if (ai, bj)  R • M(i, j)  F (false) if (ai, bj)  R • The logical matrix N representing S is given by • N(i, j)  T (true) if (bi, cj)  S • N(i, j)  F (false) if (bi, cj)  S CS1022

11. Composition with matrix (3) • Logical matrix P represents SR • If there is bk B with aiRbkandbkScj then M(i, k) = T andN(k, j)  T Since ai (SR)cj then P(i, j) is also T • Otherwise M(i, k) = F orN(k, j)  F, for all k, and P(i, j)  F • Logical matrix P representing SR is given by • P(i, j)  T (true) if M(i, k)  T andN(k, j)  T, for some k • P(i, j)  F (false) otherwise bk is the bridge between the relations that allows the product relation. CS1022

12. Matrix Boolean product A Boolean product of two matrices, where the matrices have T/F values (1/0) – It is the conjunction (and) of the disjunctions (ors), where a, b, and c reference the matrix and xi,jis the row and column of x in the matrix: etc... etc... Find a pair with a bridge between relations and mark 1/0 in result.

13. Matrix Boolean product R bill jillphilmary bill 1 0 0 0 jill 0 1 0 0 phil 1 0 1 0 mary 1 0 1 0 S bill jillphilmary bill 0 11 0 jill 0 1 1 0 phil0 0 1 1 mary 1 100 R S R S c1,2 is the same row in R, but the next column in S.

14. Composition with matrix (4) • Alternatively, P(i, j)  (M(i, 1) andN(1, j)) or (M(i, 2) andN(2, j)) ... or (M(i, n) andN(n, j)) • We write PMN For this Boolean matrix product CS1022

15. Functions (1) • Functions: special kind of (restricted) relation • Function from A to Bis a binary relation in which • Every element of A is associated with a uniquely specified element of B • That is, for all a A, there is exactly one pair (a, b) • In terms of digraphs, a function is a relation with precisely one arc leaving every element of A CS1022

16. Functions (2) Are these relations functions? Justify. • Relation x is a sibling of y on the set of People • No, since a person x may have several siblings (or none) • R = (x, x2) : x  Z (defined on Z) • Yes, since for any integer x, the value x2 is unique • S = (x, y) : x  y2 (defined on R) • No, because (2, 2) and (2, 2) belong to S CS1022

17. Terminology of functions (1) • Let f be a function from set A to set B • For each x A there exists a unique y B,(x, y)  f • We write y f(x) • We refer to f(x) as the image of x under f • We also write f : A B and it reads: • f is a function which transforms/maps each element of A to a uniquely determined element of set B CS1022

18. Terminology of functions (2) • A is the domain of f and B is the co-domain of f • The range of f is the set of images of all elements of A under f, and is denoted as f(A) f(A) f(x) : x A • Venn diagram: A B f f(A) CS1022

19. Digraphs of functions on infinite sets • When f : A B and sets A, B are infinite, we cannot draw a digraph of the function • We can plot the pairs to build a picture of f • For instance, f : R R, f(x)  x2 is as follows: y • Horizontal axis xis the domain R • Vertical axis yis the co-domain R • Curve consists of points (x, y) in the Cartesian product R R for which f(x)  x2 (2, 4) x CS1022

20. Properties of functions (1) • Let f : A B be a function • We say f is injective (or one-to-one) function if a1a2 ((a1, a2 A) and f(a1)  f(a2))  a1  a2 • That is, If output is the same then input must be the same • An injective function never repeats values • Different inputs give different outputs: a1a2 ((a1, a2 A and a1  a2)  f(a1)  f(a2)) CS1022

21. Properties of functions (2) • Let f : A B be a function • f is surjective (or onto) if its range coincides with its co-domain • For every bBthere is an a A with b  f(a) b a (bB and a A and b  f(a)) • In other words: • Each element of the co-domain is a value of the function CS1022

22. Properties of functions (3) • Let f : A B be a function • f is bijective if it is both injective and surjective CS1022

23. Properties of functions (3) • Consider the functions below (as graphs) • Not injective as f(a)  f(b)  1 • Not surjective as 2 in codomain is not a value of anyf(x) a a a 1 1 1 b b b 2 2 2 c c c 3 3 3 • Is injective as f(x) are all different • Is surjective as range and codomain are equal • Is bijective (injective and surjective) • Is injective as f(x) are all different • Not surjective as 2 in codomain is not a value of anyf(x) CS1022

24. Summary You should now know: • Inverse relations and composition of relations • Functions as a (special type) of relations • Properties of functions and some terminology CS1022

25. Further reading • R. Haggarty. “Discrete Mathematics for Computing”. Pearson Education Ltd. 2002. (Chapter 5) • Wikipedia’s entry • Wikibooks entry CS1022