Chap 2: Concept of Sets (Part 3)

 SET OPERATIONS:

DEFINITION : UNION OF SETS

Let A and B be sets. The union of the sets A and B, denoted by

A υ B, is the set that contains those elements that are either in A

or in B, or in both.

A υ B = {x | (x Є A)v (x Є B)}

DEFINITION : INTERSECTION OF SETS

Let A and B be sets. The intersection of the sets A and B, denoted by

A υ B, is the set that contains those elements that are either in A

or in B, or in both.

A υ B = {x | (x Є A) ^ (x Є B)}

DEFINITION : DISJOINT SETS (EXCLUSIVE DISJUNCTION)

Two sets are called disjoint if their intersection is the empty set.

A ^ B = Ø

DEFINITION : SET DIFFERENCES

Let A and B be sets. The difference of A and B, denoted by

A − B, is the set containing those elements that are in A but not

in B. The difference of A and B is also called the complement of

B with respect to A.

A − B = {x | (x Є A)^ (x Є B)} @ A n ~B

CHARACTERISTICS OF SET:

GENERALISED UNION OF SETS:

The union of a collection of sets is the set that contains those

elements that are members of at least one set in the collection.

We use the notation

to denote the union of the sets A1,A2, ...,An.

The Venn diagrams shows the union of the sets A, B and C

GENERALISED INTERSECTION OF SETS:

The intersection of a collection of sets is the set that contains

those elements that are members of all the sets in the collection.

We use the notation

to denote the intersection of the sets A1,A2, ...,An.

The Venn diagrams shows the intersection of the sets A, B and C

CARTESIAN PRODUCT:

Definition:

For sets A and B, their Cartesian product is

    A x B = {(a, b) | a є A ^ b є B}

* Note that AXB = BXA unless a=Ø and b=Ø

EXAMPLE

What is the Cartesian product of A = {1, 2} and B = {a, b, c}?

Solution: The Cartesian product A × B is

A × B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)}.

Definition:

The ordered n-tuple (A₁,A₂,A₃,…,A_n) is the ordered collection such that A₁ is the first element, A₂ as its second element and A_n as its nth element.

Eg. C={a,b,c,d,e,f…}

Cartesian product is when the value of a1 in (a1,a2,a3…..ai) is equal to their correspondence pair of its element, b1 in (b1,b2,b3…..bi)

It’s like:

     a1=b1

     a2=b2

     a3=b3

     ai=bi

     …………………………. So on

p/s: You can obtain the note from this link --> Chap 2: Concept of Sets (Part 3)