Chap 2: Concept of Sets (Part 2)

 SET OF NUMBERS:

SET EQUALITY:

§  Two sets A and B are equal if and only if they have the same elements.

§  We show that x⊆A and x⊆B @ ∀x(x ∈ A ↔ x ∈ B).

§  We write as A=B

§  For example:

              {1, 2, 4} = {2, 4, 1} = {1, 1, 2, 2, 2, 4, 4}

§  Those three sets are equal because they have the same elements.

VENN DIAGRAM:

§  The universal set U, which contain all the object under condition, is represented by rectangle.

§  Inside this rectangle, circle or other geometrical figures are used in order to represent sets.

For example:

    1) Draw a Venn diagram that represent S, the set of odd number less than 10.

Solution:                                                  

SUBSETS:

„  The set A is a subset of B if and only if every element of A is also an element of B.

„  We see that  A ⊆ B  only if the quantification  ∀x(x ∈ A → x ∈ B) is true.

„  To show that A is not a subset of B , we only need to find one element x ∈ A with x ∉ B OR ∃x(x∈A→x ∉B)

Proper subset:

Set A is a proper subset of B (denoted by A⊂B) if for all elements in set A is in set B, but A≠B, and there exists an element in set B which is not in set A.

@

∀x(x ∈ A → x ∈ B)^ ∃x(x∈A→x ∉B)

Showing that A is a Subset of B

To show that A ⊆ B, show that if x belongs  to A then x also belongs to B.

Showing that A is Not a Subset of B

To show that A ⊆ B, find a single

 x ∈ A such that  x ∈ B.

POWER SET:

}  Given a set S , the power set of S is the set of all subsets of the set S

}  The power set of S is denoted by P(S)

EXAMPLE

What is the power set of the set {0, 1, 2}?

Solution: The power set P({0, 1, 2}) is the set of all subsets of {0, 1, 2}. Hence,

P({0, 1, 2}) = {∅, {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2}}.

p/s: You can get the note from this link Chap 2: Concept of Sets (Part 2)