Sets by rashd
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This document defines some basic concepts in set theory: - A set is a collection of well-defined items called elements. Examples of sets are given. - The integers set Z contains all integers from negative infinity to positive infinity. The natural numbers set N contains only positive whole numbers starting from 1. - A set A is a subset of set B, written as A ⊆ B, if every element of A is also an element of B. The natural numbers set N is a subset of the integers set Z.
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3) Set operations like union, intersection, complement and difference are used to combine or relate sets according to specific rules.
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SET
A set is a well defined collection of objects, called the “elements” or “members” of the set.
A specific set can be defined in two ways-
If there are only a few elements, they can be listed individually, by writing them between curly braces ‘{ }’ and placing commas in between. E.g.- {1, 2, 3, 4, 5}
The second way of writing set is to use a property that defines elements of the set.
e.g.- {x | x is odd and 0 < x < 100}
If x is an element o set A, it can be written as ‘x A’
If x is not an element of A, it can be written as ‘x A’
Special types of sets-
Standard notations used to define some sets:
N- set of all natural numbers
Z- set of all integers
Q- set of all rational numbers
R- set of all real numbers
C- set of all complex numbers
TYPES OF SETS
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-universal set
-empty set
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-disjoint set
-cardinal set
-power set
OPERATIONS ON SET
The four basic operations are:
1. Union of Sets
2. Intersection of sets
3. Complement of the Set
4. Cartesian Product of sets
Union of two given sets is the smallest set which contains all the elements of both the sets.
A B = {x | x A or x B}
Let a and b are sets, the intersection of two sets A and B, denoted by A B is the set consisting of elements which are in A as well as in B
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If A B= , the sets are said to be disjoint.
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Sets by rashd
- 1. Set Theory By : Muhammad Jibran 17-BS(IT)-16 1
- 2. Basic Definitions • A Set is a collection of well define items, called set . • Example: • {1, 2, 3} • S = {Ali, Rahman, Rashid} 2
- 3. The Integers Set • We define the Integers to be: Z = {…, -2, -1, 0, 1, 2, 3, …} 3
- 4. Natural Numbers • Set of Natural Numbers • N = {1,2,3,4,5,6,7,8,9,……} • It is the sub set of the Whole Number set. • Whole Number set is given as • W={0,1,2.3.4.5.6.7.8……} 4
- 5. Subsets • If A and B are sets, A is called a subset of B, denoted A ⊆ B, provided every element of A is an element of B. • So, A ⊆ B means ∀x, if x ∈ A, then x ∈ B. • We also say, “A is contained in B” or “B contains A” to show this relationship. • Equivalently, we denote A ⊄ B provided ∃x ∋ x ∈ A and x ∉ B. 5
- 6. Examples of Subsets • If A = {1, 2, 3} and B = {0, 1, 2, 3, 4}, then clearly A ⊆ B. • {{1}, {2}} ⊆ {{1}, {2}, {1,2}}. • Q ⊆ R and Z ⊆ Q and N ⊆ Z. • {a, b, c} is a proper subset of {a, b, c, d}. • {a, b, c} is an improper subset of {a, b, c}. • We denote interval subsets of R as [a, b) = {x ∈ R | a ≤ x < b}. So [2, 5) ⊆ [0,5]. 6
- 7. Set Equality • We say sets A and B are equal (A = B) if every element of A is in B and every element of B is in A. • Thus, A = B means A ⊆ B and B ⊆ A. • For example {1, 2, 3} = {1, 2, 3}, but A = {1, 2, 3} ≠ {1, 2, 3, 4} = B, since 4 ∈ B but 4 ∉ A 7
- 8. Operations on Sets • Given sets A and B, which are subsets of a universal set, U, we define the following: • (Union) A ∪ B = {x ∈ U | x ∈ A or x ∈ B}. • (Intersection) A ∩ B = {x ∈ U | x ∈ A and x ∈ B}. • (Difference) A − B = {x ∈ U | x ∈ A and x ∉ B}. • (Complement) A’ = {x ∈ U | x ∉ A}. • Note that A’ = U − A. 8
- 9. Examples of Set Operations • Let U = R, A = [1, 3] and B = (2, 4). • A ∪ B = [1, 4) • A ∩ B = (2, 3] • A − B = [1, 2] • B − A = (3, 4) • Ac = (−∞, 1) ∪ (3, ∞) • Bc = (−∞, 2] ∪ [4, ∞) 9
- 10. The End 10