www.company.com module code ma1032n: logic lecture for week 4 2012-2013autumn
TRANSCRIPT
www.company.com
Module Code MA1032N:Logic
Lecture for Week 4
2012-2013Autumn
www.company.com
AgendaWeek 4 Lecture coverage:
– Sets and Subsets
– Set Listing
– Set Equality
– Special Sets
– Set Membership
– Set Builder Notation
– The empty set or null set
– Subsets
– The Universal Set
– The Cardinality of a Set
www.company.com
Sets and subsets
• A Set is any well-defined collection of objects.• "Object" and "set" are the building blocks of set theory
• The objects can be anything, are called the elements or members of the set.
• well-defined means it is always possible to determine whether a particular object is a member of the set under consideration.
www.company.com
Sets and subsets(Cont.)
• For Example:1. All the employees of a particular company,
2. The first five letters of the alphabet
3. The set of all the integers that are divisible by 5.
www.company.com
Sets and subsets(Cont.)
Representation:
Sets are usually represented by capital letters A,B,C etc.
Objects are represented by lower case letters a,b,c, etc.
www.company.com
Set Listing
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 braces (‘curly’
brackets) and placing commas in between. For example,
the set of positive odd numbers less than 10 can be
written in the following way:
1, 3, 5, 7, 9
www.company.com
Set Listing (Cont.)
If there is a clear pattern to the elements, an ellipsis (three dots) can
be used.
For example, the set of odd numbers between 0 and 50 can be
written:
1, 3, 5, 7, ..., 49 ->Finite Set
Some infinite sets can also be written in this way; for example, the set
of “all positive odd numbers” can be written:
1, 3, 5, 7, ...-> infinite Set
www.company.com
Set Listing (Cont.)
Some Examples:
1. The positive integers less than 8 ,A = 1,2,3,4,5,6,7
2. The first five letters of the alphabet B=a,b,c,d,e
3. The set of all the integers that are divisible by 5.C=5, 10,15,20,25,30,….
www.company.com
Set Builder Notation
As an alternative to set listing, a set may be described by
properties shared by all its members.
For example, suppose A = 1,2,3,4,5,6,7.
We could describe A in words by
A is the set of all positive integers less than 8.
In set builder notation this might be written
A = x : x is a positive integer and x is less than 8
www.company.com
Set builder form
Some Examples:
1. The positive integers less than 100 ,A = x: x is a positive integer less than 100
2. The letters of the alphabet B=x: x is a letter of the alphabet
3. The set of all the integers that are divisible by 5.C=x: x is a integer that is divisible by 5
www.company.com
Set Equality
Two sets A and B are called Equal if they have the same elements and to write A = B in this case.
The order in which the elements appear is not important.
If X =1,2,3 then the set A = 2,3,1 has the same elements as X and so A = X.
The sets A and B are just alternative representations of the set X.
www.company.com
Special Sets
Some sets of numbers are so important in mathematics that special symbols are reserved for them.
For Example
N is the set of all natural numbers (positive integers and zero):
N=0,1, 2, 3, 4, ...
Z is the set of all integers:
Z= ..., –3, –2, –1, 0, 1, 2, 3, ...
Z+ is the set of all positive integers:
Z+ = 1, 2, 3, ...
Q is the set of rational numbers
R is the set of real numbers.
www.company.com
Set Membership
www.company.com
Set Membership (Cont.)
www.company.com
The empty set or null set
The empty set or null set is the set containing no elements.
It is denoted by
For Example:
www.company.com
Subsets
Example:
www.company.com
Subsets (Cont.)
www.company.com
Subsets (Cont.)
www.company.com
Set Equality (Again)
www.company.com
Universal Set
When discussing a problem in set theory, the sets under
consideration are usually subsets of some fixed larger set called the
Universal Set.
For example, we might be considering subsets of the
positive integers or the real numbers R. We normally fix our
universal set at the outset of the discussion and denote it by Ω or U.
•
www.company.com
Universal Set (Cont.)
• Suppose we are considering only subsets of Ω = N
Then the set A =3,4,5,6 could be written as A = x :2 < x < 7
• However, had we been considering subsets of Ω = R then
A = x :2 < x < 7 would mean all the real numbers between 2 and 7 and would include numbers like 5/2 , 4.7, √13, π etc.
www.company.com
The Cardinality of a Set
• The number of distinct elements in a finite set A is called the
cardinality of the set, although the more transparent word size is
also used.
The cardinality of A is written |A|.
www.company.com
The Cardinality of a Set (Cont.)
• If A=a,b,c then |A|= 3
www.company.com
Operations on sets
• Intersection of two sets
• Union of two sets
• Difference of two sets
• The Complement of a Set
www.company.com
Intersection of two sets
•
www.company.com
Union of two sets
•
www.company.com
Differences of two sets
•
www.company.com
Examples
• Let A be the set of even positive integers and B the set of odd
positive integers.
www.company.com
Complement of a set
•
www.company.com
Complement of a set
•