Sets

Outline Set and Set Elements Set Representation Subset, Proper Subset, Set Equality, and Null Set

Set and Set Elements The words set and element are undefined terms of set theory just as sentence, true, and false are undefined terms of logic. A set is a collection of objects called elements or members of the set.

Set and Set Elements The elements of a set must be distinct, unordered and well-defined. It means that a set should not contain duplicates; the ordering of elements is insignificant; and you should be able to determine whether or not a certain element belongs to the set.

Set and Set Elements Examples of a Set –Letters = {a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z} –Vowels = {a, e, i, o, u} –Bits = {0, 1} –Students = {John, Don, Liza, Peter, Kathy}

Set and Set Elements The founder of set theory, Georg Cantor, suggested imagining a set as a –“collection into a whole M of definite and separate objects of our intuition or our thought. These objects are called the elements of M.” Cantor used the letter M because it is the first letter of the German word for set: Menge.

Set and Set Elements Sets are denoted by capital letters A, B, C, …, X, Y, Z. A={1, 1, 2, 3} B={1, 2, 3, 3} X={2, 3, 1, 3} Since set should not contain duplicates, the sample sets stated above can be simply written as set with three elements, {1, 2, 3}

Set and Set Elements If the elements can be counted or enumerated, then the set is said to be finite otherwise it is finite. Finite set –A={1, 1, 2, 3} –B={1, 2, 3, 3} Infinite Set - The symbol “…”,ellipsis, is a short for “and so forth.” –A={1, 2, …} –B={1, 2, 3, 3…}

Exercise Determine whether each of these sets is finite or infinite. 1.Negative integers 2.Even Integers greater than 50 3.Positive integers less than Odd integers less than 50 5.ASCII characters

Solution to Exercise Determine whether each of these sets is finite or infinite. 1.Negative integersInfinite 2.Even Integers greater than 50Infinite 3.Positive integers less than 1000 Finite 4.Odd integers less than 50 Finite 5.ASCII charactersFinite

Set Representation Tabular Form Descriptive Form Set Builder Form

Tabular Form Listing all the elements of a set, separated by commas and enclosed within braces or curly brackets{}. Example A = {1, 2, 3, 4, 5} –is the set of first five Natural Numbers. B = {2, 4, 6, 8, …, 50} –is the set of Even numbers up to 50. C = {1, 3, 5, 7, 9, …} –is the set of positive odd numbers.

Descriptive Form Stating in words the elements of a set. A = set of first five Natural Numbers. B = set of positive even integers less or equal to fifty. C = set of positive odd integers.

Set Builder Form We characterize all those elements in the set by stating the property or properties they must have to be members. Example –A = {x | x is an odd positive integer less than 10} or –A = { x| 1< x < 10 and x is a counting number}

Exercise List the members of the set 1.{x | x is a positive integer less than 100} 2.{x | x is a square of integers and x < 50} 3.{x | x is an even integer less than 100 and x can be factored by 3} 4.{x | x is an odd integer less than 50 and x is a prime number} 5.{x | x is a composite number less than 50}

Solution to Exercise List the members of the set 1.{1,2,3,…,99} 2.{1,4,9,16,25,36,49} 3.{6,12,18,24,30,36,42,48,54,60,…,98} 4.{3,5,7,11,13,17,19,23,29,31,37,41,43,47} 5.{2,4,6,8,9,10,12,14,15,16,18,20,21,22,24,25,26,27,28,30,32,33,34,35,36,38,39,40,42,44,45,46, 48,49}

Exercise Represent the following Tabular Form using Set Builder Form 1.{2, 4, 6, 8, 10, …, 50} 2.{a, e, i, o, u} 3.{MSWord, MSExcel, MSPowerpoint} 4.{Phyton, Java, C++} 5.{Fred, Dan, Tim, Stefen, Paul}

Solution to Exercise Represent the following Tabular Form using Set Builder Form 1.{x | x is an even integer less than or equal to 50} 2.{x | x is a vowel character} 3.{x | x is a Microsoft Office Application} 4.{x | x is a Programming Language} or {x | x is an Object Oriented Programming Language} 5.{x | x is a male name} or {x | x is a boy name}

Set Representation Not all sets can be represented using both methods. For instance, suppose you’re to represent the set of all real numbers between 0 and 1. It’s impossible to list down all the elements of this set, and thus you can only use the rule method to describe this set.

Set Representation N – Set of Natural Numbers –N = {1, 2, 3, … } W – Set of Whole Numbers –W = {0, 1, 2, 3, … } Z – Set of Integers –Z = {…, -3, -2, -1, 0, +1, +2, +3, …} –Z = {0, ±1, ± 2, ± 3, …} –{“Z” stands for the first letter of the German word for integer: Zahlen.}

Set Representation E – Set of Even Integers –E = {0, ±2, ± 4, ± 6, …} O – Set of Odd Integers –O = {±1, ±3, ±5, …} P – Set of Prime Numbers –P = {2, 3, 5, 7, 11, 13, 17, 19, …} Q – Set of Rational Numbers (or Quotient of Integers) –Q = {x | x = p/q ; p, q  Z, q  0}

Set Representation Q – Set of Irrational Numbers –Q’ = {x | x is not rational numbers} R – Set of Real Numbers C – Set of Complex Numbers

Set Representation Superscript symbols such as +, -,  represent positive, negative and non negative(includes zero) values respectively. Example: – Z  denotes the set of all non-negative integers. –Z + denotes the set of all positive integers

Subset If A and B are two sets, A is called a subset of B, written A  B, if, and only if, any element of A is also an element of B. Symbolically A  B   x (x  A  x  B) For all elements of x, if x is an element of A then x is an element of B.

Subset The phrases A is contained in B and B contains A are alternative ways of saying that A is a subset of B. When A  B, then B is called a superset of A. When A is not subset of B, then there exist at least one x  A such that x  B. –A  B ⇔ ∃ x such that x ∈ A and x ∈ B. Every set is a subset of itself.

Example Let –A = {1, 3, 5} –B = {1, 2, 3, 4, 5} –C = {1, 2, 3, 4} –D = {3, 1, 5} Then –A  B ( Because every element of A is in B ) –C  B ( Because every element of C is also an element of B )

Example –A  D ( Because every element of A is also an element of D and also note that every element of D is in A so D  A ) –A  C ( Because there is an element 5 of A which is not in C )

Examples Let –A = {1} –B = {{1}} –C = { 1, {1} } A  C B  C A  B since 1  {{1}} Consider the set symbol as a box { }. Thus {1} is “a box holding 1”, while {{1}} is “a box holding a box holding {1}”. The only subsets of {1} are {1} and { }; the subsets of {{1}} are {{1}} and { }.

Exercise Suppose that A = {2, 4, 6}, B = {2, 6}, C = {4, 6}, and D = {4, 6, 8}. Determine which of these sets are subsets of which other of these sets.

Solution to Exercise Suppose that A = {2, 4, 6}, B = {2, 6}, C = {4, 6}, and D = {4, 6, 8}. Determine which of these sets are subsets of which other of these sets. –B  A –C  D –C  A

Proper Subset Let A and B be sets. A is a proper subset of B, if, and only if, every element of A is in B but there is at least one element of B that is not in A, and is denoted as A  B. Example Let –A = {1, 3, 5} –B = {1, 2, 3, 5} then A  B ( Because there is an element 2 of B which is not in A).

Set Equality Two sets A and B are equal if, and only if, every element of A is in B and every element of B is in A and is denoted A = B. Symbolically: A = B iff A  B and B  A

Set Equality Example Let –A = {1, 2, 3, 6} –B = the set of positive divisors of 6 –C = {3, 1, 6, 2} –D = {1, 2, 2, 3, 6, 6, 6} Then A, B, C, and D are all equal sets.

Null Set A set which contains no element is called a null set, or an empty set or a void set. It is denoted by the Greek letter  (phi) or { }. Example –B = {x | x 2 = 4, x is odd number} =  (there does not exist any odd number whose square is 4) –Z + = {x | x < 0 } =  ( there does not exist any positive integer whose values is less than 0)  is regarded as a subset of every set

Exercise Determine whether each of the following statements is true or false. 1. x  {x} 2. {x}  {x} 3. {x}  {x} 4.   {x} 5.   {x}

Solution to Exercise Determine whether each of the following statements is true or false. 1. x  {x}True 2. {x}  {x}True 3. {x}  {x}False 4.   {x}True 5.   {x}False

Summary A set is a collection of objects called elements or members of the set. The elements of a set must be distinct, unordered and well-defined If the elements can be counted or enumerated, then the set is said to be finite otherwise it is finite. Set can be represented using Tabular Form, Descriptive Form, and Set Builder Form

Summary Subset states that If A and B are two sets, A is called a subset of B, written A  B, if, and only if, any element of A is also an element of B. A is a proper subset of B, if, and only if, every element of A is in B but there is at least one element of B that is not in A, and is denoted as A  B. Two sets A and B are equal if, and only if, every element of A is in B and every element of B is in A and is denoted A = B.

Summary A set which contains no element is called a null set, or an empty set or a void set. It is denoted by the Greek letter  (phi) or { }.

Summary The Universal Set is represented by the interior of a rectangle, and the other sets are represented by circle within the rectangle.