SETS

• Everywhere, at home, in school, or in the market, the word set is already a byword.

• Whenever we talk of a collection of appliances, books, flowers, players, scholars, kitchen utensils, heroes, and the like, we are using the concept of set loosely.

• In mathematics, we also talk of sets such as sets of numbers, set of geometric figures, set of geometric theorems, and solution sets. In this chapter, we shall consider the mathematical definition of sets and the different operations on sets.

SETS

• A set is a well-defined collection of distinct objects referred to as members or elements of the set.

• For example, we have the set of planets, the set of continents in the world, the set of colors in the rainbow, the set of even numbers, the set of real numbers, the set of factors of 36, and so on.

• Sets are represented by capital letters and their elements are enclosed by braces { }.

• Set Notations• Sets can be described in two ways, namely,

the roster or listing method and the rule method. Obviously, the roster method is done by enumerating the elements of the set.

• For example, we have • A = {1, 2, 3, 4, 5}, • V = {a, e, i, o, u}, • C = {1, 2, 3, …}. • The three dots in C, called ellipsis, are used to

indicate that the list is to continue indefinitely to include other numbers following the same pattern shown in the first few terms.

• The rule method for describing sets is done by stating either a rule, a definition or a condition that will make it clear that an object belongs to the set. Sometimes, it is written in the "set-builder" notation , that is,

• A= {xx is a natural number • less than 6}

• read as "set A of all x’s such that x", where x is the variable, the vertical bar () replaces the words "such that" and the given set A is called the replacement set of the variable x.

• The variable x is to be replaced by a name of an element of the set. Any lower case (small) letter may be used as the variable. • In case the elements of a set are

letters, these are written in lower case.

Roster Method Rule Method Set-Builder Notation

1. C = {2, 4, 6, 8}

C is the set of even counting numbers less than 10.

C = {x|x is aneven countingnumber lessthan 10}

2. M = {m, o, n, d, a, y}

M is the set of letters in the word Monday

M = {x|x is a letter in the word Monday}

3. O = {1, 3, 9} O is the set of odd divisors of 36.

O = {x|x is an odd divisor of 36}

• To indicate that an element belongs to a given set, the Greek letter epsilon, , is used.

• For example, 8 C (read 8 is an element of C or 8 belongs to C), m M, and 9 O.

• However, 3 C (read 3 is not an element of C), x M, and 5 O.

• The number of elements in a set A is called the cardinal number or cardinality of A and this is denoted by n(A). • For example, n(D) = 7 and n(M) = 12,

where D is the set of days of the week and M is the set of months of the year.

Drill Exercises• A. Give the cardinality of each of the following sets and

enumerate its elements:• 1. The lady presidents of the Philippines • 2. The sense organs• 3. The suits of a deck of cards• 4. The Philippine coins in the new millennium• B. Give two sets with exactly: • 1. 1 element• 2. 2 elements• 3. 3 elements

• C. Using the roster method, write the elements of the set • 1. E of all even prime numbers.• 2. M of all multiples of 5 less than 30.• 3. D of all positive composite divisors of 54.• 4. E of all even factors of 36.• 5. O of all odd divisors of 24.• D. Describe each of the following sets:• 1. M = {January, March, May, July, August, • October, December}• 2. N = {3, 6, 9, 12, 15, 18}• 3. O = {1, 2, 3, 4, 5, 6}• 4. P = { a, b, c, d, e, f, g, h, i, j}• 5. Q = {2}

• Use the set builder notation to describe the sets D above.

• Use the roster method to specify the following sets:

• A = {aa is a counting number less than 8}• B = {bb is a positive divisor of 48}• C = {cc is a positive multiple of 12 less than 50}• D = {dd is common factor of 42 and 96}• E = {ee is a positive number less than 100 that is

divisible by 7}

Kinds of Sets• Sets are classified as follows:• Empty set or null set is a set without any

element or member. • For example, the set of odd numbers divisible

by 2 is an empty set. This is denoted by { } or and its cardinality is 0.

• Unit set is a set with only one element. • For example, the set of months with

less than 30 days consists of only one element which is February.

• Finite set is a set that contains a definite number of elements.

• Thus, a finite set contains a countable number of elements as in

• 1. the set of factors of 54, • 2. the set of students in a certain classroom,

or• 3. the set of consonants in the English

alphabet.

• Infinite set is a set that does not contain a countable number of elements.

• As in the case of the set of integers.• Universal set is the totality of all the elements

under discussion and it is denoted by the capital letter U.

• For example, if V is the set of vowels and C is the set of consonants, then U is set of all the letters in the English alphabet, that is,

• U = {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}.

Relationships Between Pairs of Sets

• Two or more sets may be classified as follows:• Disjoint sets are sets which have no element

in common. • Sets V and C , where V is the set of vowels

and C is the set of consonants, are disjoint sets.

• Equal sets are sets that have exactly the same elements, regardless of the order the elements are listed.

• For example, • if X = {a, b, c, d, e} • and Y = {b, c, a, e, d}, then X = Y.

• Equivalent sets are sets that have the same cardinality, that is, the elements of one set can be matched, one-to-one, with the elements of another set.

• For example, sets • M = {a, b, c, d, e} • and N = {1, 2, 3, 4, 5} are equivalent sets.

• We can associate each element of M with an element of N as shown below.

• M = {1, 2, 3, 4, 5}

• N = {a, b, c, d, e}

• The double-headed arrows indicate one possible matching between the elements of sets M and N.

• Note that n(M) = 5 and n(N) = 5. • Symbolically, we write M N.

EXERCISES

• 1. Are equal sets equivalent? • 2. Are two equivalent sets equal?. • 3. Are two empty sets

equivalent? equal?

Subsets

• Set A is a subset of set B if for every element of A is an element of B.

• In symbols, we write • A B (read as "A is a subset of B").

• We say that A B if and only if for every x A, x B. • If A B, we may also write • B A, and say that A is contained in B

or B contains A or B is a superset of A.

• If A B and A B, we say that A is a proper subset of B. • The symbol A B means that every

element of A belongs to B and B contains at least one element not found in A.

• We say that A B if and only if for every x A, x B, there exists some y B such that y A.

• For example, if • A = {1, 2, 3, 4}, • B = {1, 2, 3, 4, 5} and • C = {1, 2, 3, 4, 5, 6, 7}, • then A C, A B and B C.

• On the other hand, • if A B and B A and that A = B, then A is an

improper subset of B and B is an improper subset of A. In symbols, we write

• A B and B A.

• Now consider M = {1, 2, 3}, then N = {1, 2} is a subset of M.• How many subsets has M? • Notice that it is possible to have subsets of 1

element, 2 elements, and 3 elements, that is, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3},

• {1, 2, 3}.

• In fact a set is a subset of itself.• Also, the empty set { } is a subset of M since

there is no element of an empty set which is not an element of M.

• The empty set is a subset of every set. • The empty set and the set itself are called

trivial subsets of any given set.

• How many subsets does a set have?• This can be answered by listing the

subsets of a set with 1 element, 2 elements, 3 elements, and so on and then observing patterns.

Sets Number of Elements

Number of Subsets

A = {a} 1 2

B = {a, b} 2 4

C = {a, b, c}D = {a, b, c, d}

34

816

• Observe that the total number of subsets of any given set may be determined by the formula 2n, where n is the cardinality or the number of elements of the set.

• Since n(A) = 1, the number of subsets of A is 21 = 2.

• The number of subsets of B is 22 = 4 since n(B) = 2.

• Similarly, the number of subsets of C is 23 = 8 since n(C) = 3, while the number of subsets of

• D is 24 = 16 since n(D) = 4.

Exercises

• A. Indicate the kind of sets described in each of the following:

• 1. the positive integers divisible by 10• 2. the three-digit natural numbers• 3. the even prime numbers • 4. the seats in a certain movie house• 5. the qualified voters in a certain precinct

• Given: A = {1, 2, 3, 4}. • Write T if the statement is true and F if the

statement is false.• 1. 2 A • 2. A A• 3. A • 4. {4} A• 5. {2, 3} A

• 6. 1 A• 7. A • 8. 1, 4 A• 9. A A • 10. 5 A

OPERATIONS AND RELATIONS ON SETS

• The relations between and among sets can be represented pictorially by using the Venn diagram which was introduced by James Venn, an English logician.

• A Venn diagram usually consists of a rectangle which represents the universal set U and closed curves drawn within the rectangle to represent subsets of U.

• The rectangle represents the universal set U and

• the circle represents the subset A with elements written inside

• the circle. • Notice that x A and a U, • but a U and a A. • Therefore, A U.

• Consider the following sets:• U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, • A = {1, 2, 4, 6, 8}, and • B = { 1, 3, 4, 5, 7}. • Let us draw the Venn diagram that will

represent the relationship among these sets.

• Notice that the elements common to sets A and B are found in the region where the circles overlap while the elements that belong to the universal set U, but not to set A or set B are found inside the rectangle but outside the circles.

Operations on Sets

• Consider two sets A and B which are subsets of the universal set U.

• The union of two sets A and B, denoted by A B (read as "A union B"), is the set of all elements found in A or in B or in both A and B.

• In symbols, we write • A B = {xx A or x B}. • For example, if • A = {1, 2, 3, 4, 5} and • B = {2, 4, 6, 8}. then A B = {1, 2, 3, 4, 5, 6, 8}.

• The intersection of two sets A and B, denoted by A B

• (read as "A intersection B"), is the set of all elements common to both A and B.

• In symbols, we write • A B = {xx A and x B}.

• Using the preceding example, we see that A B = {2, 4}.

• The complement of a set A, denoted by A’ (read as "A complement" or "A prime"), is the set of all elements in U that are not found in A, that is,

• A’ = {xx U and x A}.

• With A = {1, 2, 3, 4, 5} and • U = {1, 2, 3,4 5, 6, 7, 8, 9}, • A’ = {6, 7, 8, 9}. • What is A A’ ? A A’?

• The difference of two sets A and B, denoted by A – B (read as "A minus B"), is the set of all elements found in A but not in B. In symbols, we write

• A – B = {xx A and x B}.

• Thus, if A = {1, 2, 3, 4, 5} and • B = {2, 4, 6, 7, 8, 9}, then • A – B = {1, 3, 5} and • B – A = {6, 7, 8, 9}.

• The cross product of two sets A and B, denoted by A B (read as “A cross B”), is the set of ordered pairs (x, y) such that x A and y B. Every element of A is paired with every element of B in that particular order.

• If A = {1, 2} and • B = {3, 4}, then • A B = {(1, 3), (1, 4), • (2, 3), (2, 4)}• and • B A = {(3, 1), (3, 2), • (4, 1), (4, 2)}