SETS

Empty Set (null set):

A set is a collection of well-defined objects called elements.

some denotations

Finite – Infinite sets

UNIVERSAL SET (U)

SUBSETS

Georg Cantor is the first mathematician who uses the “SET CONCEPT” in mathematics. image

Sets are denoted by capital letters, such as A, B, C....

x 
A

n(A)

x 
A

Representation of Sets

  1. Venn diagram
  1. Listed form
  1. Set-builder form

Elements of sets are written inside of the set brackets { } by putting comma between two elements.

Elements of a set are written in a simple closed shape.

image

The set that contains all the elements being discussed is called universal set, and it is denoted by U. image

A set which contains no element is called empty set denoted by symbol { } or

A = {1, 3, 5, 7, 9, 11}

N = { 0 , 1 , 2 , 3 , 4, ... }

Equal Sets:

A=B;

A⊆ B

image

AB={xxA  xB }

Properties of subsets

b) The empty set is a subset of any set. That is Ø.......A.

c) Every set is a subset of itself , that is for any set A , A.....A.

a) Every set is a subset of universal set , that is for any set A , A .....U.

d) If A⊂B and B⊂C then A....C

e) If A⊂B and B⊂A then A ......B.

Number of subsets

2n(A) – 1 = 2k – 1

2
n(A)=2k

OPERATIONS ON SETS

UNION OF SETS

Remark

INTERSECTION OF SETS

image

image

Ekran Görüntüsü (9799)

The properties of Union and Intersection

c) Associative Property

d) Distributive property

b) Commutative Property

e) Subset property

a) Idempotent property

Ekran Görüntüsü (9801)

Ekran Görüntüsü (9802)

Ekran Görüntüsü (9803)

Ekran Görüntüsü (9804)

Ekran Görüntüsü (9806)

DIFFERENCE OF TWO SETS

The difference of the sets A and B is the set of elements of A which does not belong to B, and it is
denoted by A−B or A\B.

COMPLEMENT OF A SET

Ekran Görüntüsü (9809)

Properties of Complement:

Ekran Görüntüsü (9811)

De Morgan’s Law

Ekran Görüntüsü (9813)

Common Properties of Complement and Difference of the Sets

Ekran Görüntüsü (9815)

SET PROBLEMS

data for set problems

Ekran Görüntüsü (9818)

ordered pairs

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CARTESIAN PRODUCT (Cross Product)

Ekran Görüntüsü (9820)