4. 1. What reminds you with our activity this day?
2. How do you find the activity?
3. What do you think is our topic for today?
5. BASIC CONCEPT OF SET THEORY
P R E P A R E D B Y : J o a n G u e r r e r o
6. OBJECTIVES:
define the basic concept of set theory;
01.
reflect on the importance of the set
theory connected to our life and;
02.
provide and solve a problem related to
theorem through group activity.
03.
7. SET
is a collection of objects called
elements.
can be FINITE and INFINITE.
The objects in a set are called the
elements or members of the set.
8. To designate a set, we use the
notation {x: P(x)},
P(x) is a one-variable open
sentence description of the
property that defines the set.
Example: set A= {1,3,5,7,9,11,13} may
be written as {x: x ϵ N, x is odd, and x <
14}.
Set A is a subset of a set B if every
element of A is also an element of B,
and write A⊂B or B⊃A.
Capital letters will be used to denote
sets and lowercase letters to denote
the elements in sets.
If a is an element of set A, we write
a∈A. If a is not an element of a set A,
we write a∉A
For instance, to specify the fact that a
set A contains four elements a,b,c,d,
we write
A={a,b,c,d}
10. Theorem 2.1.1
(a) For every set A, ∅ ⊆A.
(b)For every set A, A ⊆A.
(c)For all sets A, B, and C,
if A⊆B and B⊆C, then A⊆C.
11. Theorem 2.1.2
If A and B are sets with no
elements, then A = B.
PROOF:
• Since A has no elements, the sentence
(x ∈ A⇒ x ∈ B) is true
• Therefore, A ⊆ B. Similarly, (x ∈ B⇒ x ∈ A) is true,
so B ⊆A.
• Therefore, by definition of set equality, A=B.
12. Theorem 2.1.3
For any sets A and B, if A ⊆B
and A ≠ ∅, then B ≠ ∅.
PROOF:
• Suppose A ⊆B and A ≠∅. Since A is nonempty,
there is an object t such that t∈A.
• Since t ∈ A. Since t ∈ A, t ∈ B. Therefore, B ≠∅.
13. Theorem 2.1.4
If A is a set with n elements,
then P(A) is a set with 2^n
elements.
PROOF:
• If n=0, that is, if A is the empty set, then P(∅)=
{∅}, which is a set with 2^0 = 1 elements.
• Thus the theorem is true for n=0.
14. Theorem 2.1.5
Let A and B be sets. Then
A⊆B if and only if P(A) ⊆
P(B).
PROOF:
16. Instruction:
In a long bond paper, look for problem involving theorems through internet or
books. Solve the problems, show your solution and explain how the theorems was
applied. (ONE PROBLEM FOR EACH THEOREM ONLY)
Theorem 2.1.1
Theorem 2.1.2
Theorem 2.1.3
Theorem 2.1.4
Theorem 2.1.5
17. Assignment:
In a whole sheet of paper kindly write a reflection on what did you learn for
today’s topic and how was our topic important to our life? Minimum of 300 words
and maximum of 500 words.
