Mathematics in the Modern World: Chapter Three

Language of Sets

1. LANGUAGE OF SETS Presentation By Group 2

2. LIST OF CONTENTS Defining set Methods of Describing sets classification of sets RELATION OF SETS OPERATIONS ON SETS Properties of sets

3. SETS -are well-defined collections of distinct objects represented by capital letters, with elements separated by commas.

4. For example, the set of natural numbers can be represented as A={1,2,3,…}

5. METHODS OF DESCRIBING SETS

6. -the set are enumerated and separated by comma. ROSTER METHOD

7. For example, the set of vowels in the English alphabet can be represented as V={a,e,i,o,u}

8. -it is used to describe the elements or members of the set. 2. SET-BUILDER NOTATION

9. For example, the set of all x such that x is a natural number less than 5 can be expressed as B={x|x∈N,x<5}

10. CLASSIFICATION OF SETS

11. Finite has a countable number of elements, while an Infinite set has an uncountable number of elements . FINITE AND INFINITE SETS

12. Example: The set of days in a week. FINITE SET D= {Monday,Tuesday,Wednesday,Thursday, Friday,Saturday,Sunday}

13. Example: The set of all natural numbers. INFINITE SET E= {1,2,3,4,…}

14. The empty set, denoted by ∅ or {}, contains no elements. EMPTY SET

15. Example: The set of all unicorns. EMPTY SET F=∅

16. It is the set that contains all possible elements within a particular context, often denoted by U. UNIVERSAL SET

17. Example: If we are discussing the set of all integers, the universal set could be represented as U={…,−3,−2,−1,0,1,2,3,…} UNIVERSAL SET

18. A set A is a subset of set B if all elements of A are also elements of B, denoted as A⊆B. SUBSETS

19. Example: If G={1,2,3}, then H={1,2} is a subset of G (i.e., H⊆G). SUBSETS

20. RELATION OF SETS

21. Two sets are equal if they contain exactly the same elements, denoted as A=B. EQUAL SETS

22. Example: Let I={1,2,3} and J={3,2,1} . Then I=J . EQUAL SETS

23. Two sets are equivalent if they have the same number of elements, regardless of the actual elements themselves. EQUIVALENT SETS

24. Example: Let K={a,b,c} and L={1,2,3} . Both sets have the same number of elements, but they are not equal. EQUIVALENT SETS

25. The intersection of two sets A and B, denoted as A∩B, consists of elements that are common to both sets. JOINT SETS

26. Example: Let M={1,2,3,4} and N={3,4,5,6} . The intersection is: M∩N={3,4} JOINT SETS

27. Two sets are disjoint if they have no elements in common, meaning A∩B=∅. DISJOINT SETS

28. Example: Let O={1,2} and P={3,4} . These sets are disjoint because they have no elements in common: O∩P=∅ DISJOINT SETS

29. OPERATIONS OF SETS

30. The union of two sets A and B, denoted as A∪B, is the set of elements that are in A, in B, or in both. UNION

31. Example: Let Q={1,2,3} and R={3,4,5} . The union is: Q∪R={1,2,3,4,5} UNION

32. The intersection of two sets A and B, denoted as A∩B, is the set of elements that are in both A and B. INTERSECTION

33. Example: Using the same sets Q and R : Q∩R={3} INTERSECTION

34. The difference between two sets A and B, denoted as A−B or A∖B, is the set of elements that are in A but not in B. DIFFERENCE

35. Example: The difference between Q and R : Q−R={1,2} DIFFERENCE

36. The complement of a set A, denoted as A' , consists of all elements in the universal set U that are not in A. COMPLEMENT

37. Example: If the universal set U ={1,2,3,4,5} and Q={1,2,3} , then the complement of Q is: Q′=U−Q={4,5} COMPLEMENT

38. PROPERTIES OF SETS

39. It states that the order in which elements are combined does not affect the result. AUB=BUA A∩B=B∩A COMMUTATIVE PROPERTY

40. AUB = BUA A∩B = B∩A COMMUTATIVE PROPERTY

41. It states that when combining sets, the way in which they are grouped does not affect the result. ASSOCIATIVE PROPERTY

42. ( A ∪ B ) ∩ C = A∩ ( B ∩ C) ASSOCIATIVE PROPERTY

43. It describes how union and intersection interact with each other. DISTRIBUTIVE PROPERTY

44. A∪(B∩C) = (A∩B)∩(A∪C) A∩(B∪C) = (A∩B)∪(A∩C) DISTRIBUTIVE PROPERTY

45. The union of a set with the empty set is the set itself, and the intersection of a set with the universal set is the set itself. IDENTITY PROPERTY

46. A∪Ø = A A∩U = A IDENTITY PROPERTY

47. It is the set that includes all the elements of the universal set that are not present in the given set. COMPLEMENT PROPERTY

48. A∪A' = U COMPLEMENT PROPERTY

49. The union of any set with the same set will result in the set itself. IDEMPOTENT PROPERTY

50. A∩A = A A∪A = A IDEMPOTENT PROPERTY

51. THANK YOU