SET THEORY

BASIC CONCEPTS IN SET THEORYDefinition: A set is a collection of well-defined

objects, called elementsExamples: The following are examples of setsa. The months of the year that begins with letter Jb. The set of engineering drawing instrumentsc. The set of PB teams in this year’s PBA series. Names of sets are designated by using capital

letters such as A, B… To indicate that an element belongs to a given set, the symbol epsilon є is used and the symbol epsilon slash for an element that does not belong to the set.

Example• “4 is an element of the set S” (4 є S)• “1 is not an element of the set S (1 є S)• What are the elements of the set S = {blue,

orange yellow, black}? (There are four elements in the S namely blue,

orange yellow and black)

Methods of Describing A Set• Listing or Roster Method – list the elements, separate

them by commas and enclose this list in a pair of braces, { }

Describe the given sets by roster methoda. If A is the set of the days of the week that begin

with the letter S, then A = {Saturday, Sunday}b. If B is the set of letters in the word love, then B = {l,

o, v, e}c. The set of digits that make up the telephone

number 715-51-70 {0, 1, 5, 7}Note: the elements within the set are never repeated

and the elements can appear in any order

Methods of Describing A Set• Set in the Set-Builder Notation{x| x is a natural number between 2 and 10} =

{x|x = 3, 4, 5, 6, 7, 8, 9}

Kinds of Sets• A set A is said to be a subset of another set B if

every element of A is also elements of B. In symbol, A ⊆ B.

• Proper Subsets – a set A is said to be a proper subset of another set B and there is at least one element in set B not in set A, then a is a proper subset of B. In symbols A⊂ B.

Examples

Given the sets A = {1, 2, 3}, B = {1, 2, 3, 4, 5, 6, 7}, C = {3, 5, 7}, D = {3, 2, 1}1. A ⊆ B? True since all the elements of A are

also in B.2. A⊂ B? True since set A is a subset of Set B

and the elements of set B namely 4, 5, 6, 7 are not found in set A.

3. A ⊄ C? True since not all the elements of a are contain in C.

4. C ⊆ D? False since not all the elements of C are contained in D

Example5. A ⊆ D. True, since all of the elements of A

are contained in D6. D ⊆ A. True, since all of the elements of D

are contained in A

Kinds of Sets• Equal Sets – Set A is said to be equal to set B,

written A = B, if and only if A ⊆ B and B ⊆ A.• Equivalent Sets – Set A is said to be equivalent

to set B if and only if the two sets have exactly the same cardinality of set.

• Null Set – A set which does not contain any element is called a null or empty set. It is denoted by the symbol ⌀. The symbol { } can also be used to denote an empty set.

Kinds of Sets• Finite and Infinite Set. The set whose

elements can be counted is called a finite set, while an infinite set is a set whose elements cannot be determined.

Example. The set A = {1, 2, 3] is finite set since it contains 3 elements, while B = {x|x is a natural number divisible by 2} is an infinite set since the number of elements in this set cannot be determined. B = {2, 4, 6, …}

The number of subsets of a finite set A is 2n(A)

Operations on the Sets• Universal Sets. The totality of elements under

consideration as the elements of any set. The symbol is ∪.

• The union of sets A and B, denoted by A ∪ B is defined as the set whose elements are in A or in B or in both A and B. In symbol, A ∪ B = {x |x є A or x є B}

• Common Set (∩) – The elements of Set A which are also elements of Set B.

Operations on the Sets• Relative Complement. The relative

complement of set B with respect to A designated as A – B is defined as: A – B = {x|x є B and X ∉ B}

• Absolute Complement. The absolute complement of A designated by A’ is the set of elements of the universal set ∪ which do not belong to A, that is A’ = {x|x є ∪ and x ∉ A}

Operations on the Sets• Ordered pair. If x and y represent two objects

(alike or different) with x specified as the first object, then the symbol (x,y) is called an ordered pair.

Examples• Given the following sets A = {1, 2, 3), B = {2, 3,

4, 5}, C = {4, 5}, and D = {2, 3, 7] form the following sets;

1. A ∩ B = {1, 2, 3} ∩ {2, 3, 4, 5} = {2, 3}2. B ∩ C = {2, 3, 4, 5} ∩ {4, 5} = {4, 5} = C3. B – A = {2, 3, 4, 5} – {1, 2, 3} = {4, 5}Given A = {1,2} and B {a, b, c}4. A x B = {(1,a), (1,b), (1,c), (2,a) (2,b) (2,c)}5. B x A = {(a,1), (a,2), (b,1), (b,2), (c,1), (c,2)}

Homework

A. Write each set by listing the elements.1. The even integers between 1 and 16.2. {y|y is a natural number greater than 9}3. The months with 30 days4. The letters in the word college5. {z|z is a whole number and a multiple of 5}

B. Use mathematical symbols to write the following statements.

1. A is a subset of G2. The number 7 is not an element of a null set3. The set of even prime numbers4. The number 3 is an element of T5. The null set is a subset of A

Homework (Cont.)

C. Enclose within braces a list of the elements of each set.1. The natural numbers less than 7.2. The even natural numbers between 2 and 133. The months of the year that begins with letter J4. The first four natural numbers multiplied by 35. The square of the first 4 natural numbers.D. Use the descriptive method to designate each of the

following sets.1. {3,6,9,12}2. {even natural numbers}3. {odd natural numbers}4. {1, 8, 27, 64}5. {March, May}

Homework (Cont.)E. Tell if the two sets are equal, if they are

equivalent, if one is the subset of the other, or if none of these is true.

1. A = {4,7,11} B = {11, 7, 4}2. A = {a, e, i, o, u}, B = {u, v, x, y}3. A = {0}, B = ⌀4. A = 5, B = {3, 4}5. A = {earth, venus, pluto}, B = {0, 1, 2}

Homework (Cont.)

F. Given A = {a}, B = {a, b}, C = {b, c, d}, find1. A ∩ B2. B ∪ C3. C x C4. A x B5. B - C