Sets Section 2.1 Basic Notions of Sets Section 2.2 Operations with sets Section 2.3 Indexed Sets

1. SetsSection 2.1 Basic Notions of Sets Section 2.2 Operations with sets Section 2.3 Indexed Sets The most fundamental notion in all of mathematics is that of a set. We say that a set is a specified collection of objects, called elements (or members) of the set. We denote sets by capital letters A, B, … and elements by lower case letters, like x, y , … and so on. If anelement xbelongs to a set A, we denote this by x A, if not wewrite x A. Instructor: Hayk Melikya melikyan@nccu.edu

2. Specifying set • There are various ways to specify a set. For the set of natural numbers less than or equal to 5, you could write {1, 2, 3, 4, 5}. • For sets that cannot be specified by a list, we describe the elements by some property common to the elements in the set but no others, such as in the description A = { x | P (x)} which reads “the set of all xsuch that2 the condition P(x) is true.”

3. Common Sets:

4. Subsets • We say that a set A is a subset of a set B if every element of A is also an element of B. • Symbolically, we write this as A  Band is read “A is contained in B.” • Finally, the notation A  Bmeans that A is not a subset of B. • Sets are often illustrated by Venn diagrams, where sets are represented as circles and elements of the set are points inside the circle.

5. Equality of Sets: • Two sets are equal (A = B) if they consist of exactly the same elements. In other words, they are equal if (A = B) if and only if (x) (( xA  x B) or (A = B) iff (x) (( xA  x B)  (xB x A )) another way: (A = B) if and only if (A  B B  A). Empty Set:The set with no elements is called the empty set (or null set) and denoted by the Greek letter (or sometimes the empty bracket { } )

6. Theorem 1 (Guaranteed Subset) For any set A, we have  A. Proof : Since the goal is to show x  x  A our job is done before we begin. The reason being that the hypothesis x  of the implication is false, being that  contains no elements, hence the proposition is true regardless of the set A. In other words  is a subset of any set. END

7. Theorem 2 (Transitive Subsets) Let A, B and C be sets. If A  B and B  C then A  C . Proof: We will prove the conclusion A  Cand use the hypothesis as needed. Letting x  Athe goal is to show x  C. Since x  Aand using the assumption A  B, we know x  B . But the second hypothesis says B  C, and so we know x  C. Hence, we have proved A  C , which proves the theorem. END

8. Subset and Membership:

9. Power Set P(A) An important set in mathematics is the power set. For every set A, we denote by P(A)the set of all subsets of A. Theorem 3 (Power Set) Let A and Bbe sets. Then A  Bif and only if P (A)  P(B). Proof: (A B)  (P(A)  P(B)): We start by letting X  P(A) and show X  P(B)(and use A  Bas our “helper”). Letting X  P(A)we have X Aand hence X B. But this means X P(B) and so we have shown P(A)  P(B). (P(A)  P(B)) (A  B) : We let x  A and show x  B. If x  A,then {x }  P(A), and since P(A) P(B)we know {x } P(B). But this means x Band so A  B.

10. Sec 2.2 Operations on SetsUnion, Intersection and Complement • In traditional arithmetic and algebra, we carry out the binary operations of + and × on numbers. In logic, we have the analogous binary operations of  and  on sentences. In set theory we have the binary operations of union  and intersection  of sets, which in a sense are analogous to the ones in arithmetic and sentential logic. Definition ( Union): The union of two sets A and B, denoted A B, is the set of elements that belong to A or B or both. Symbolically A B = {x | x  A  x  B }

11. Definition ( Intersection): The intersection of two sets A and B, denoted A  B, is the set of elements that belong to A and B. Symbolically A  B = {x | x  A  x  B }

12. Definition( Complement): The compliment of A, denoted Acis the set ofelements belonging to the universal set Ubut not A. Symbolically Ac = {x | x ÎU  x  A } .

13. Definition (Relative Complement or Difference): The relative complement of A in B, denoted, B \ A, is the set of elements in B but not in A. Symbolically B \ A = {x | x  B x  A } The concepts of union, intersection and relative complement of sets can be illustrated graphically by use of Venn diagrams. Each Venn diagram begins with an oval representing the universal set, a set that contains all elements of in discussion. Then, each set in the discussion is represented by a circle, where elements belonging tomore than one set are placed in sections where circles overlap.

14. Venn diagrams for two overlapping sets.

22. Section 2.3

23. Definitions:

24. Example1

26. Solution:

27. Example2:

28. Extended Laws: