1. www.company.com Module Code MA1032N: Logic Lecture for Week 4 2012-2013Autumn

2. www.company.com Agenda Week 4 Lecture coverage: –Sets and Subsets –Set Listing –Set Equality –Special Sets –Set Membership –Set Builder Notation –The empty set or null set –Subsets –The Universal Set –The Cardinality of a Set

3. www.company.com Sets and subsets A Set is any well-defined collection of objects. "Object" and "set" are the building blocks of set theory The objects can be anything, are called the elements or members of the set. well-defined means it is always possible to determine whether a particular object is a member of the set under consideration.

4. www.company.com Sets and subsets (Cont.) For Example: 1.All the employees of a particular company, 2.The first five letters of the alphabet 3.The set of all the integers that are divisible by 5.

5. www.company.com Sets and subsets (Cont.) Representation: Sets are usually represented by capital letters A,B,C etc. Objects are represented by lower case letters a,b,c, etc.

6. www.company.com Set Listing A specific set can be defined in two ways. If there are only a few elements, they can be listed individually, by writing them between braces (‘curly’ brackets) and placing commas in between. For example, the set of positive odd numbers less than 10 can be written in the following way: {1, 3, 5, 7, 9}

7. www.company.com Set Listing (Cont.) If there is a clear pattern to the elements, an ellipsis (three dots) can be used. For example, the set of odd numbers between 0 and 50 can be written: {1, 3, 5, 7,..., 49} ->Finite Set Some infinite sets can also be written in this way; for example, the set of “all positive odd numbers” can be written: {1, 3, 5, 7,...}-> infinite Set

8. www.company.com Set Listing (Cont.) Some Examples: 1. The positive integers less than 8, A = {1,2,3,4,5,6,7 } 2. The first five letters of the alphabet B={a,b,c,d,e} 3. The set of all the integers that are divisible by 5. C={5, 10,15,20,25,30,….}

9. www.company.com Set Builder Notation As an alternative to set listing, a set may be described by properties shared by all its members. For example, suppose A = {1,2,3,4,5,6,7}. We could describe A in words by A is the set of all positive integers less than 8. In set builder notation this might be written A = {x : x is a positive integer and x is less than 8}

10. www.company.com Set builder form Some Examples: 1. The positive integers less than 100, A = {x: x is a positive integer less than 100 } 2. The letters of the alphabet B={x: x is a letter of the alphabet} 3. The set of all the integers that are divisible by 5. C={x: x is a integer that is divisible by 5 }

11. www.company.com Set Equality Two sets A and B are called Equal if they have the same elements and to write A = B in this case. The order in which the elements appear is not important. If X ={1,2,3} then the set A = {2,3,1} has the same elements as X and so A = X. The sets A and B are just alternative representations of the set X.

12. www.company.com Special Sets Some sets of numbers are so important in mathematics that special symbols are reserved for them. For Example N is the set of all natural numbers (positive integers and zero): N={0,1, 2, 3, 4,...} Z is the set of all integers: Z= {..., –3, –2, –1, 0, 1, 2, 3,...} Z + is the set of all positive integers: Z + = {1, 2, 3,...} Q is the set of rational numbers R is the set of real numbers.

13. www.company.com Set Membership

14. www.company.com Set Membership (Cont.)

15. www.company.com The empty set or null set The empty set or null set is the set containing no elements. It is denoted by For Example:

16. www.company.com Subsets Example:

17. www.company.com Subsets (Cont.)

18. www.company.com Subsets (Cont.)

19. www.company.com Set Equality (Again)

20. www.company.com Universal Set When discussing a problem in set theory, the sets under consideration are usually subsets of some fixed larger set called the Universal Set. For example, we might be considering subsets of the positive integers or the real numbers R. We normally fix our universal set at the outset of the discussion and denote it by Ω or U.

21. www.company.com Universal Set (Cont.) Suppose we are considering only subsets of Ω = N Then the set A ={3,4,5,6} could be written as A = {x :2 < x < 7} However, had we been considering subsets of Ω = R then A = {x :2 < x < 7} would mean all the real numbers between 2 and 7 and would include numbers like 5/2, 4.7, √13, π etc.

22. www.company.com The Cardinality of a Set The number of distinct elements in a finite set A is called the cardinality of the set, although the more transparent word size is also used. The cardinality of A is written |A|.

23. www.company.com The Cardinality of a Set (Cont.) If A={a,b,c} then |A|= 3

24. www.company.com Operations on sets Intersection of two sets Union of two sets Difference of two sets The Complement of a Set

25. www.company.com Intersection of two sets

26. www.company.com Union of two sets

27. www.company.com Differences of two sets

28. www.company.com Examples Let A be the set of even positive integers and B the set of odd positive integers.

29. www.company.com Complement of a set

30. www.company.com Complement of a set