The Joy of Sets. Joshua Knowles. Introduction to Maths for Computer Science, School of Computer Science University of Birmingham

Transcription

1 School of Computer Science University of Birmingham Introduction to Maths for Computer Science, 2015

2 Definition of a Set A set is a collection of distinct things. We generally use upper case letters to denote a set. To indicate the members of a set, we can list them explicitly within { }, e.g. A = {1, 2, 3, 4, 5} B = {a, b, c, z} C = {man, house, cat, } Notice: The following is not a set as its members are not distinct. {1, 1, 1, 1, 2, 2, 2, 2} These kinds of collection are called multi-sets, but we will not deal further with them here.

3 Elements, membership and cardinality The members of a set are called members or elements. Given the set A = {1, 2, 3, 4, 5} we write 1 A, which is read as 1 is an element of A. Similarly we write 7 A. The cardinality of a set C is the number of elements it has, and is denoted as C. For the above set A, we can see that A = 5.

4 Defining sets by the properties of their elements Another way to define a particular set is to specify the properties that the members of the set have. This is obviously useful for large sets or sets with infinite members. Here is an example of such a definition: B = {x is a natural number x > 5, x/2 is a natural number}. The first part tells us what larger set our elements are drawn from (here the natural numbers), also called the universal set U. The symbol is read as such that, and then a list of properties are given separated by commas or by logical connectives (and, or) to define the subset of U that B is. What set does B denote above?

5 Some Useful Sets We give the important sets their own special (reserved) symbols. Here are some of the most useful ones. is the empty set, U or U is the universal set, A is the complement of A. That is, everything in the universal set not in A, N is the set of natural numbers, Z is the set of integers, Q is the set of rational numbers and R is the set of real numbers. Exercise: Using these symbols, define the set of positive irrational numbers less than 100.

6 Subsets and Supersets Consider two sets A = {3, 4, 5, 6, 7} and B = {4, 5, 6}. Then it is true that B A, and B A. (B is a proper subset of A, and B is a subset of A). It is also the case that A B, and A B. (A is a proper superset of B, and A is a superset of B).

7 The Power Set The power set of a set is very useful. It is the set of all subsets of a set, and is usually denoted by P. If then A = {1, 2, 3} P(A) = {{ }, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}. You will notice that P(A) = 2 A.

8 Logical and existential symbols It is useful in defining sets to be able to call on some logical symbols. x is read as for all x. x is read as there exists an x such that... stands for logical AND. For example: x > 2 x odd. stands for OR. max(a) is the greatest element of A, assuming A is an ordered set. min(a) is the minimum element of A, assuming A is an ordered set.

9 Exercises Use symbols from the previous page to define these. (i) The natural numbers co-prime with 2 and 5. (ii) Given a set B of natural numbers, the set A of numbers such that for every element x of B, there is a corresponding element y of A where y = x + 5, and no others. (Is this well-defined?) (iii) The natural numbers except for the range

10 Intersection and Union A B means the set consisting of elements in A or B, where the or is inclusive (i.e. the elements can be in both A and B.) This is read as A union B. Similarly, A B means the set consisting of elements in A and B (i.e. the elements must be in both A and B.) This is read as A intersection B. Show that (A B) (A B) A B.

11 Relationships between the numbers Complete the following N R =? N Q =? P(R) =? N Z =? (the star denotes non-negative) {log 1 0(y), y {1, 5, 10, 50, 100, 500, 1000,...}} =?

12 Venn Diagrams Notice that by drawing the two sets with an intersection, the most general relationship has been shown. (The intersection may be empty or not.)

13 Venn Diagrams A general representation of three sets (subsets of a Universal set). Any of the subregions may be empty.

14 Men Venn Diagram

15 Tuples Sometimes we want to reason about complex objects. Let us say we want to talk about the students taking this maths class. We could describe each of them by the following tuple: (name, number, birthday, gender). Call the set of all such tuples, T. The set, this maths class, could be defined as C = {x T x refers to a person who comes to Intro2Maths}. How could we describe the set of names of members of this class who are older than 20?

16 Answer: V = {x C x.birthday prior to 9 November 1995} assuming that all individuals in the class are unique when described by the tuple: (name, number, birthday, gender). If they are not, potentially our set V could be smaller than the real set of people older than 20 in the class! Why?

17 More on Venn Diagrams Try drawing the following relationships A B (Don t forget the Universal set) A B (A B) C A B

18 Laws The following laws apply to sets. These are similar to the laws governing multiplication and addition in arithmetic. Let A, B, C be subsets of a Universal set U. 1. (A B) C = A (B C) 2. (A B) C = A (B C) 1. & 2. are the associative laws. 3. A B = B A 4. A B = B A 3. and 4. are the commutative laws. 5. A (B C) = (A B) (A C) 6. A (B C) = (A B) (A C) 5. and 6. are the distributive laws. Slide based on material from p.63, K. Devlin (2004) Sets, Functions and Logic, 3rd Ed.

19 Laws (continued) Let A, B, C be subsets of a Universal set U. 7. (A B) = A B 8. (A B) = A B 7. and 8. are called the De Morgan laws. 9. A A = U 10. A A = 9. and 10. are called the complementation laws. 11. (A ) = A 11. is the self-inverse. How would you prove laws 1 to 11? Slide based on material from p.63, K. Devlin (2004) Sets, Functions and Logic, 3rd Ed.

20 Proof of the distributive law (6) We need to prove A (B C) = (A B) (A C). Define the left hand side L = A (B C) and the right hand side R = (A B) (A C). First prove that L R. We do this by considering an element x L and proving it is necessarily in R too. (Do this yourself) We then do the analogous proof for L R. We consider an element x R and prove it is necessarily in L too. (Do this yourself) Having established the two results L R and R L, we have shown L = R and the proof is complete. Slide based on material from p.65, K. Devlin (2004) Sets, Functions and Logic, 3rd Ed.

21 Proving the Self-Inverse Law (11) We need to show that (A ) = A. We can prove this by rewriting this in terms of the difference of sets. A is everything in U not in A, which can be written as U A. The complement of A is everything in U not in A. This can be written as U A. So, now we can rewrite our original statement as U A = U (U A). The right hand side is A (from the correspondence between set difference and normal arithmetic difference).