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 Sets

Section 2.1 Basic Notions of Sets

Section 2.2 Operations with sets

Section 2.3 Indexed Sets 

Instructor: Hayk Melikya melikyan@nccu.edu

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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 an element x belongs to a set A, we denote this by x A, if not we write x A.

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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 x such that2 the condition P(x) is true.”

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Common Sets:

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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 B and is read “A is contained in B.”

Finally, the notation A B means 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.

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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) (( xA x B) or

(A = B) iff (x) (( xA x B) (xB 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 { } )

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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, beingthat contains no elements, hence the proposition is true

regardless of theset A. In other words is a subset of any set.

END

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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 C and use the hypothesis as needed.

Letting x A the goal is to show x C . Since x A and 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

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Subset and Membership:

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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 B be sets. Then A B if 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

B as our “helper”). Letting X P(A) we have X A and 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 B and so A B .

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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.

SymbolicallyA B = {x | x A x B }

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Definition ( Intersection):

The intersection of two sets A and B, denoted A B , is the set

of elements that belong to A and B.

SymbolicallyA B = {x | x A x B }

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Definition( Complement):

The compliment of A, denoted Ac is the set of elements belonging to the universal set U but not A.

Symbolically Ac = {x | x U x A } .

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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 to more than one set are placed in sections where circles overlap.

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Venn diagrams for two overlapping sets.

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Section 2.3

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Definitions:

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Example1

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Solution:

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Example2:

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Extended Laws: