Thursday, May 7, 2015

Ch. 1. Sets - Concept Review

1,1 Set Explanation

Mathematicians encountered server difficulties in defining set. They realized that there is a need for some undefined (primitive) terms. Thus set and elements are undefined terms in mathematics.

Set is synonymous with the words, ‘collection’, aggregate’, ‘class’, and is comprised of elements.

The words ‘element’, ‘object’, and ‘member’ are synonymous.

Sets designated by specific letters.

N: natural numbers
Z : integers
Z+: positive integers
Q: rational numbers
Q+: positive rational numbers
R: real numbers
R+: positive real numbers
C: complex numbers

1.2 Description of a set

Sets can be described by roster method or set-builder method.

1.3 Types of sets

Empty set
Singleton set
Finite set
Infinite set
Equivalent set
Equal set
Universal set
Power set

1.4 Theorems on subsets

1. Every set is a subset of itself.
2. The empty set is a subset of every set.
3. The total number of subsets of a finite set containing n elements is 2ⁿ

1.5 Universal Set

A set that contains all sets in a given context (in simple terms in the given problem) is called the universal set.

1.6 Power Set

Given a set A, the collection of all subsets of A is called the power set of A and is denoted by P(A).

1.7 Venn Diagrams

In Venn diagrams the universal set U is represented by points within a rectangle and its subsets are represented by points in closed curves (usually circles) within the the rectangle. If two sets A and B have some common elements, they will be shown two intersecting circles. If the two sets are disjointed sets, then they are represented by two non-intersecting circles. If set B is a subset of A,  then B is shown as a circle inside the circle representing set A.

1.8 Operations on Sets

Union of Sets

Intersection of Sets

Difference of Sets

Symmetric Difference of Two Sets

Disjoint Sets

Complement of a Set

1.9 Laws of algebra of sets

1. Idempotent laws

(i) A U A = A
(ii) A ∩ A = A

2. Identity laws

(i) A U ф = A
(ii) A ∩ U = A

3. commutative law
(i) A U B = B U A
(ii) A ∩ B = B ∩ A

4. Associative laws

(i) (A U B) U C = A U (B U C)
(ii) (A ∩ B) ∩ C = A ∩ (B ∩ C)

5. Distributive laws

(i) A U (B ∩ C) = (A U B) ∩ (A U C)
(ii) A ∩ (B U C) = (A ∩ B) U (A ∩ C)

6. De-morgan’s laws

(i) (A U B)’ = A’ ∩ B’
(ii) (A ∩ B)’ = A’ U B’

1.10 Some more theorems

Theorem 1

If A and B are two sets

(i) A – B = A ∩ B’
(ii) B – A = B ∩ A’
(iii) A – B = A  A ∩ B = ф
(iv) (A – B) U B = A U B
(v) (A-B) ∩ B = ф
(vi) A is a sub set of B  B’ is a subset of A’
(vii) (A-B) U (B-A) = (A U B) – (A ∩ B)

Theorem 2

If A, B and C are three sets, then

(i) A – (B ∩ C) = (A-B) U (A-C)
(ii) A – (B U C) = (A-B) ∩ (A-C)
(iii) A ∩ (B-C) = (A ∩ B) - (A ∩ C)
(iv) A ∩ (B Δ C) = (A∩B) Δ (A∩C)

1.11 Number of Elements in Sets

Note union operation and universal set have the same symbol in these pages. Hence identify appropriately.

n(A) denotes the number of elements in the set A. Similarly n(B) and n(C).

If A,B and C are finite sets. U is the finite universal set, then

(i) n(A U B) = n(A) +n(B) – n(A∩B)

(ii) n(A U B) = n(A) +n(B)  implies A, B are disjoint non-void sets.

(iii) n(A-B) = n(A) –n(A∩B)

(iv) n(A ΔB) = Number of elements which belong to exactly one of A or B
= n((A-B) U (B-A))

(v) n(A U B U C) = n(A) + n(B) + n(C) – n(A∩B) – n(B∩C) – n(A∩C)+n(A∩B∩C)

(vi) No. Of elements in exactly two of the sets A,B,C
= n(A∩B) + n(B∩C)+n(C∩A)-3n(A∩B∩C)

(vii) No. Of elements in exactly one of the sets A,B,C
= n(A) +n(B)+n(C)-2n(A∩B)-2n(B∩C)-2n(A∩C)+3n(A∩B∩C)

(viii) n(A’ U B’) = n((A∩B)’) = n(U) – n(A∩B)

(ix) n(A’∩B’) = n((AUB)’) = n(U)-n(A∩B)

Reference: Objective Mathematics by R.D. Sharma, Dhanpat Rai & Sons

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