Sets, Functions and Relations

1. Sets, Functions and Relations CSC 2110 Tutorial

2. Self introduction • You can call me Isaac • I’m responsible for the tutorials of the first 3 weeks and the first classwork • If you have questions, you may email me at wsfung@cse.cuhk.edu.hk • Or come to SHB 115

3. What is a set? • Q: Give me one element of each of the following sets? The set of English letters, the set of English words that starts with ‘d’ and ends with ‘e’, the set of all natural numbers, the set of all 8 digit telephone numbers, the set of factors of 30030, the set of integers { x N|100<x<120, } • The order of elements does not matter e.g. {a, b, c} is the same set as {b, a, c} Q: If A = (Jennifer, Ken, John, May) is a sequence of people who are ordered by their ages, is A just the same as the set {Jennifer, Ken, John, May}? • The set {1, 2, 3, 3} is same as the set {1, 2, 3} Q: If I want to record the number of times my friends visit my home, can I do this by just adding his/her name into a set every time he/she visits me?

4. What can be in a set? • A set may contain infinitely many elements e.g. the set of real numbers Q: Give me another set that has infinitely many elements • A set can also contain zero element e.g. the empty set, := { } Q: Can you give me another set that has zero elements? • Types of elements doesn’t matter, e.g. S = {11/13, red, ‘CSC2110’, (10,10) } • A set can also be an element of some other set e.g. X = {{1}, {1,2}, {1,2,3}} Suppose A={1}, B={2}, C={3} Q: Is {A, B, C} the same as {1, 2, 3}? How many elements are there in the set { { {1, 2} } }? What about A {A} X?

5. How can we specify a set? • We can specify a set by • listing all the members of the set, e.g. {1, 2, 3} Q: Could you list all the elements of the set of integers? • stating the properties of the set members, e.g. X = {x Z | x is even} Q: Try specifying the sets of Fibonacci numbers (take home exercise) • the results of set operations on some other sets e.g. A is the set of all quadrilaterals whose four sides have equal length (rhombus), B is the set of quadrilaterals which have two adjacent angles equal to (trapezium), C is the set of quadrilaterals such that the 2 pairs of opposite sides are parallel, so what is , and ?

6. How can we specify a set? Q: What is the complement of the positive even integers? (if the universe is 1. positive integers, 2. even integers, 3. integers) Q: If X has been defined to be the set of right-angled triangles and Y is the set of isosceles triangles, you are asked to specify the set of all right-angled isosceles triangles, which method would you prefer to use? Q: Suppose M={a,b,c,d,e}, N={b,d}, P={c,e}, Q={b,c} Express {a,e} in terms of these 4 sets using only basic set operations

7. Venn Diagram • When we work with just 2 or 3 sets, it is often useful to draw the Venn diagram • Suppose the blue circle represents a set A and the red circle represents a set B • Try to find the regions corresponding to the complement of A, A B, A B, A\B • Try to derive the De Morgan’s law and

8. Venn Diagram, continued • Suppose the red circle represents the set of multiples of 4, the blue circle represents the multiples of 15 and the yellow circle represents the multiples of 10. • Try to figure out the meaning of each region • Try to derive the distributive laws ,

9. Subsets of a set • e.g. the set of prime numbers is a subset of the natural numbers, the set of core courses is a subset of all the courses, a set is a subset of itself, the empty set = { } is a subset of any set, the intersection of two sets A and B is always a subset of A and B, A and B are always subsets of the union of A and B Q: Is the set of even numbers a subset of the composite numbers? Q: Let x and y be two integers. If F is the set of factors of the largest common factor of x and y, is F a subset of the union of the set of factors of x and the set of factor of y? Q: Let A={1,2,3,4,5,6}, B={1,2,3,4,6}, C={1,3,4,5,6}, D={3,4} Give me a subset of A that is not a subset of B and C but not contains D as its subset

10. Subsets of a set, cont. • X = Y (X and Y contain the same elements) if and only if X Y and X Y (can you see why?) e.g. the set of multiples of 10 equals the intersection of the set of multiples of 2 and 5 Q: Let x and y be two integers If s is the smallest common multiples of x and y, does the set of factors of s equal the union of the set of factors of x and the set of factors of y? • The power set of a set X, Pow(X) is the set of all the subsets of X e.g. Let X={1, 2, 3}. Pow(x)={ ,{1}, {2}, {3}, {1,2}, {2,3}, {3,1}, {1,2,3}} • Q: Give me the power set of the power set of { 0, 1 } • Q: Give me a set whose power set has only one element

11. What is a function? • e.g. the identity function, f(x):=x e.g. the set membership function of a set X, e.g. is a function whose domain and codomain are sets of functions e.g. let x be a student ID, f(x):=name of the student who has this student ID e.g. currency conversion formula, suppose x is the price of something in $HK, f(x):=the value of x in $US f image of x x domain codomain

12. What is not a function? g • e.g. g(x):=1/x is not a total function if the domain is , as 1/0 is undefined Q: Is f(x):=log(x) a total function if the domain is the set of real numbers larger than 0? • e.g. Define f(x):=y if y^2=x f(x) is not a function as 2^2=(-2)^2=4, the element 4 has two images 2 and -2 under f • e.g. Let X be a set, f(X):=an element of X, f(X) is not a functon as X may have no elements or X can have more than one elements x What is g(x)? domain codomain X is not mapped to some element in the codomain h y x h(x)=y or h(x)=z? z domain codomain X is mapped to two elements in the codomain

13. Surjective functions • Roughly speaking, if a function is surjective, then each element in the codomain will have AT LEAST one arrow pointing to it e.g. f(x):=sin(x), domain = , codomain = [-1, 1] f(x) is surjective Q: Is f(x) still surjective if the codomain is ? e.g. f(x):=1, domain = , codomain = {1} f(x) is surjective but it is not surjective if we add anything other than 1 to its codomain e.g. Suppose f(x):=x+1 and the codomain is the set of even numbers Q: If f(x) is surjective, what should be the domain of f(x) Q: If there are more elements in the codomain than in the domain, can this function be surjective?

14. Injective functions • If a function is injective, then each element in the codomain can have AT MOST one arrow pointing to it e.g. f(x):=course code of course x, domain=set of courses, codomain=set of course code f(x) is injective as no two courses share one course code e.g. f(x):=cos(x), codomain=[-1, 1] f(x) is not injective if the domain is but f(x) is injective if the domain is Q: If a function is injective, can it be true that there are more elements in the domain than in the codomain?

15. Bijection and inverse • A function f is a bijection if it is total, surjective and injective e.g. f(x):=x+1 is a bijection between the set of even numbers and the set of odd numbers e.g. f(x):=-x is a bijection between the set of positive numbers and the set of negative numbers • If there is a bijection between 2 sets A and B, the sizes of A and B are the same e.g. We can construct a bijection between the set of English letters and the set {1,2,…,26} to count the number of letters e.g. We can construct a bijection between the set of natural numbers and the set of rational numbers to count the size of a infinite set Q: Try constructing a bijection between the set of natural numbers and the set of positive rational numbers • If we reverse the direction of the arrows in a bijection, we get a new function, which is called the inverse of the original function. Q: What is the inverse of f(x):=(x-2)^3? Q: Does f(x):=(x-2)^2 have an inverse?

16. Composite functions • A composite function is a function formed by cascading 2 functions • e.g. f(x) := (sin(x))^2 can be viewed as the composite of two functions h(y) and g(x) where h(y) := y^2 and g(x) := sin(x) • When we write , actually we means When we want to evaluate f(x) we just pass the input x to g and then pass the output of g as the input of h, and finally we return the output of h as the output of f

17. Composite functions, cont. h g Q: Is f(x) a total function if both g(x) and h(y) are total functions? Assume f(x), g(x), h(y) are all total functions. Is f(x) bijective if both g(x) and h(x) are bijective? Are both g(x) and h(x) bijective if f(x) is bijective? • Graphically, we may join the graphs of the functions g and h to form the graph representing f f g h x g(x) f(x)=h(g(x))

18. Functions vs Relations • In a function, each element in the domain is associated with one element in the codomain • What if we want to associate each student with the course he/she has taken? A student may have taken >1 course • One approach is to model this by a function whose domain is the set of students and the codomain is the set of all possible combinations of courses (notice that the set of combinations of courses can be much larger than the set of all courses) • Besides, the elements in the codomain are sets of courses. However what we want to model is the relationship between students and courses instead of relationship between students and set of courses 1130, 1500, 2100 Student A 2100, 2510, 3150 Student B Student C 1500, 3150, 3160

19. Functions vs Relations, cont. • It is more natural to associate the students with the courses they take • This requires us to allow each student to be associated with more than one courses • We call such a mapping a relation Using this approach, it is much easier to answer questions like: Who have taken 2100? Are there any courses taken by both students A and B? Are there any students who have taken both 2100 and 3150? 1130 Student A 1500 2100 Student B 2510 3150 Student C 3160

20. Some special relations • Here we only consider binary relations - relations between two objects • You should have seen many binary relations before and many of them describe relations between 2 elements of the same set • e.g. a = b, a < b, “equal to” and “smaller than” are relations between pairs of real numbers • e.g. P  Q “if and only if” are relation between pairs of propositions • e.g. Alice “is a relative of” Bob is a relation between two people • e.g. John “is a friend of” Mary is also a relation between two people

21. Graphs of relations • When a binary relation is defined between elements of the same set, we can use another type of diagram to represent this relation • In this type of diagram, we have only one set of points representing elements of the set. If two elements (x, y) are in the relation, we draw an arrow pointing from x to y (notice that the order matters, e.g. 2>1, the converse 1>2 is not true) • e.g. 1 2 The “is a friend of” relation among some people The “defeats” relation among some football teams

22. 1 2 3 4 5 1 2 3 4 The “equals” relation on the natural numbers 1, 2, 3, 4, 5 The “larger than” relation on the integers 1, 2, 3, 4 More examples 3 4 5 Each point represents a person, and the arrows corresponds to the “has the same surname” relation Can you observe some of the properties of diagrams 2, 3, 4? Can you tell what do they have in common with diagram 5?

23. Equivalence relations • Notice that the elements in the diagram are divided into some disjoint subsets. Elements in the same subset have arrows pointing to each other (and themselves) but there are no edges crossing from one subset to another subset • There are many relations which have diagrams similar to the diagram in the last example • e.g. The “is similar to” relation on triangles, all equilateral triangles are similar • e.g. The “equals” relation on rational numbers, 2/3 = 4/6 • e.g. The “has the same remainder when divided by 7” relation on integers, 3 mod 7 = 10 mod 7 Q: How many disjoint subsets are there in this relation? • We call these relations Equivalence relations

24. Partition of a set • If two sets A and B do not share any common elements, i.e. , we say that they are disjoint • Suppose X1, X2, … , Xn are subsets of a set X. If • Their union is equal to X, and • Every pair of them are disjoint Then we say that X1, X2, … , Xn form a partition of X • Refer back to the diagram in example 5. If the relation is a equivalence relation, we can form a partition by the following procedures • Let each element form a subset which contains only this element • Whenever there is an arrow pointing from an element x to an element y, combine the subset containing x and the subset containing y • Continue until there is no arrow crossing two subsets • The resulting collection of subsets is a partition of the set • This partition has the properties highlighted in the last slide • We call a subset in this partition, an equivalence class e.g. The even and odd numbers form two equivalence classes under the relation “having the same remainder when divided by 2”

25. Symmetry, Transitivity and Reflexivity • You may observe that there are some properties that are shared by example 5 and examples 2, 3 and 4 • In examples 2 and 5, whenever there is an arrow pointing from an element x to an element y, then there is an arrow pointing from y to x. We say that such a relation is a symmetric relation • In examples 3 and 5, for any three elements x, y and z, whenever there is an arrow pointing from x to y and an arrow from y to z, then there must be an arrow from x to z We say that this relation is a transitive relation • In examples 4 and 5, every element in the set has an arrow pointing from itself to itself We say that this relation is a reflexive relation • In fact, a relation is a equivalence relation if and only if it is symmetric, transitive and reflexive

26. Which of these relations are symmetric, reflexive or/and transitive? x < y, x ≤ y (x, y are numbers) X Y, X and Y are disjoint (X,Y are sets) A is married to B (A, B are people) p is orthogonal to q (p, q are straight lines) P  Q, P -> Q (p, q are propositions) Someone can travel from x to y by walking and taking lift but not leaving a building (x, y are rooms) X and Y star in the same film (X, Y are actors/actresses) X is the ancestors of Y (X, Y are people) x and y do not have common factors (x, y are integers) Are the following relations equivalence relations? If yes, what are the equivalence classes? x and y have the same age/sex (x and y are people) There are lectures of x and y on the same day of the week (x, y are courses) x and y are partners in the same project group (x, y are students taking CSC2110) x and y are married (x, y are people) x and y are sibling (brother or sister) of each other (x, y are people) More relations

27. Tips and feedback • For each concept mentioned in this tutorial, try to find your own examples • The diagram representations (the Venn diagram, the diagrams of functions/relations) are usually more concrete and easy to understand • We will discuss the classwork next week Take a look at it first and ask me next time if you have questions • Some topics like the club & strangers problem, the halting problem, uncountability of real numbers may be a bit more difficult, let me know if you want more explanation