MODULE 4: The Language of Sets

Sets and Set Notations

• WELL DEFINED SETS

1. The set of female presidents of the Philippines (Direct, clear)

• NOT WELL DEFINED SETS

1. The set of good Filipino writers (Indirect, unclear, writers of?)

Rules in writing sets

1. Name the set using capital letter of English Alphabent (A,B,C..Z)

2. The objects inside the set are called elements. We write them in small letters (a, b, c, …z)

3. The Elements are separated by commas (,), and are written inside braces { }.

• EXAMPLES:

• A = {f, r,e, s, h, m, n}

symbol ∈

• We use this if an object is an element of the set.

• EX. A={f,r,e,s,h,m,n}

1. f is an element of set A, or f is a member of set A

2. f is in A, or f belongs to A, therefore

3. f ∈ A

• On the other hand, if an object is not an element of the set,we use the symbol ∉ as in a is not an element of set A, a ∉ A

THE SET OF REAL NUMBERS

• Natural Numbers (N) - Counting numbers/positive integers. Example: 1,2,3…

• Integers (Z) - Natural numbers/positive, negative, zero (0) Examples: -3, -2, -1, 0, 1, 2 3

• Rational Numbers (Q) - Numbers that can be represented in fraction (a/b) where a and b are integers and b≠0 (undefined). If written in decimal, it is terminating (.33333=.33…)and repeating Examples: -15, -2, 0, 23, -1/4, 3/7, 15/2, -2.75, 1.625, -0.333...

• Irrational Numbers (Q’) - Numbers that cannot be expressed as the ratio of two integers. They are the non- repeating and non-terminating decimals. Examples: √2=1.414213562..., π=3.141592654...

• Real Numbers (R) - All rational and Irrational Examples: 1/4, 3/4, 5/6, and π, square root of 5

Enumeration or Roster Method

• The elements of a set are listed or enumerated in any order but no repetition and are enclosed by curly braces.

• You will just copy the statement but in a set form. Example statement: E is the set of consonants in the word decency. roster: E = {c, d, n, y}

Rule Method

• The members of the set are defined by stating their common properties. Like you will put the statement into a set form but instead, you will now use symbols.

• {x I x > -2} or {x : x >-2}. This is read as“the set of all x’s, such that x is greater than-2”. The ”I” is read as ”such that” The “:” can also be used instead of ”I”.

• Example statement: E is the set of consonants in the word decency. rule: E = {xIx is a consonant in the word decency}

• NOT WRITTEN IN PLURAL, MUST BE SINGULAR

Finite Sets

• A set whose elements are limited or countable. The cardinality of a set is the number of its elements. The cardinality of set A is denoted by n(A) or IAl

EXAMPLE:

A= {xIx is a positive integer less than 17}

A= {1,2,3, ...,16}.

Cardinality n(A) =16

Infinite Sets

• A set whose elements are unlimited or cannot be counted. the infinity of the set is denoted by three dots (...) called an ellipsis and is read "and so forth.”

EXAMPLE:

W= {0, 1, 2, 3, ...}

n(W) = Infinity

Unit Set or Singleton Set

• A set with only one element

EXAMPLE:

F = { xIx is an integer greater than10 but less than 12}

F = {11}

n(F) =1

Empty Set or Null Set

• A set with no object or element . The symbols "{ }" and "∅ " are used to denote an empty set.

EXAMPLE:

H= { xIx is an integer greater than-1 but less than 0}

H= { } or H=∅

n(H) =0

Equal Sets

• Two sets containing the same elements are equal. Symbolically, A = B.

• The symbol "≠" is used to denote that two sets are not equal.

EXAMPLE: Let,

A={13, 14, 15,16}

B = {14, 15}

D= { xIx is a natural number greater than12 but less than 17}.

E = {15, 16, 14, 13}

from the given set,

A = D = E, and

A≠B, D≠B, and E≠B.

Equivalent Sets

• If Sets A and B have the same number of elements or the same cardinality, then they are equivalent sets.

• They may have different elements or some common elements only. We use the symbols “~” or “≡” to denote that sets are equivalent.

EXAMPLE: Let

A={11,12, 13}

B = {x, y, z}

C = {λ, β, α}.

From the given sets,

A~B, A~C, and B~C. Because all contains 3 elements

Disjoint Sets or Non-Intersection

• Two sets with no common elements are disjoint

• EXAMPLE: LET

• R={2, 4, 6},

• S = {11, 13, 15}

• Then, R and S are disjoint.

• The set of wholenumbersand theset of negativenumbers aredisjoint.

Subsets

• A is a subset of B if every element of A is in B. The phrases "A is contained in B" and "B contains A" are other ways of saying that A is a subset of B. Symbolically, A ⊆ B

• The number of subsets of a given set with "n" elements can be computed using the formula 2ⁿ

• Note: Every set is a subset of itself. The empty set is a subset of every set.

EXAMPLE: Consider the sets:

J = {5, 6, 7}

K=[8,7, 6,5}

L = {5, 6, 7, 8, 9}

then, we can say that:

J⊆K, J ⊆L, and K⊆L.

Using the solution for solving the n:

Given: M={7, 8, 9} 2ⁿ

n(M) = 3, therefore,

2³ = 2×2×2 = 8

Proper Subset

• If every element of A is in B, but there is at least one element in B that is not in A, then A is a proper subset of B. In symbols, we have A⊂B.

• If a set has "n "elements, the number of proper subsets can be verified using 2ⁿ – 1.

• Every set is not a proper subset of itself. A null set is a proper subset of every set.

EXAMPLE: Consider the sets:

J = {7, 6, 5}

K={8,7, 6,5}

L = {9, 8, 7, 6, 5}

We can say that,

J ⊂ K, J ⊂ L, and K ⊂ L.

Using the solution for solving the n

Given: J = {7, 6, 5}

For the number of proper subsets of J:

2ⁿ – 1 = 2³ - 1

=8- 1

=7

Superset

• If A is a subset of B, then we can say that B is a superset of A, denoted by B ⊇ A.

EXAMPLE:

Consider thesets:

J = {2, 4, 6}

K = {2, 4, 6, 8}

L = {2, 4, 6, 8, 10}

Then, we can say that

K ⊇ J, L ⊇ J, and L ⊇ K

Power Sets

• The set of all subsets of a set A is called the power set of A, denoted as P(A).

• The cardinality (or the number of elements) of the power set of A is given by |P(A)| = 2ⁿ

EXAMPLE:

set A is called the power set of A, denoted asP(A).

The cardinality (or the number of elements) of the power set of A is given by |P(A)|=2n.

Let N= {q, r, s}

n(N) = 3

|P(N)| = 2³ =8

P(N) = {∅ , {q}, {r}, {s}, {q, r}, {q, s}, {r, s}, {q, r, s} }

Universal Set

• The universal set is the set containing all the possible elements under consideration or the set containing all the elements of its subsets, without repetition.

EXAMPLE: Let

A={a, c, e,g}

B={b, d, f}

C={a, d, g, h}, then

U={a, b, c, d, e,f,g,h}

Complementary Sets

• Sets A and B are complementary if they have no common elements, and their union is the universal set.

• In symbol, the complement of A is Aᶜ or A′

EXAMPLE:

Let U= {10, 11, 12, 13,14,15,16, 17}

A = {10, 12, 14}

B = {11, 13, 15, 17}

C = {10, 12, 13,14,16} then,

A’ ={11, 13, 15,16,17}

B’ = {10, 12, 14, 16}

C’ = {11, 15, 17}

Union

• The union of two sets A and B is the set containing all the elements of both sets.

• In symbols, we write A ∪ B.

EXAMPLE: Let

A = {1, 3, 5}

B = {1, 5, 8, 10}

C = (3, 6, 12, 14}

D = { } then,

A U B = {1, 3, 5, 8, 10}

B U C = {1,3,5,6,8,10,12,14}

C U D= {3, 6, 12, 14}

Intersection

• The intersection of two sets A and B is the set containing the common elements of both sets.

• In symbols, we write A ∩ B.

EXAMPLE: Let

A = {1, 3, 5}

B = {1, 5, 8, 10}

C = (3, 6, 12, 14}

D = { }, then

A ∩ B = {1, 5}

B ∩ C = { } or ∅

C ∩ D= { } or ∅

Cross Product of Sets

• The cross product of two non- empty sets A and B is the set of all ordered pairs (a, b), where the first element a is from set A, and the second element b is from set B.

• The symbol "x" is used for the operation. Read A x B as A cross B. A x B ≠ B x A

EXAMPLE: Let

A = {5, 7, 9},

B = {r, s, t}

Find: A x B

Solution:

A x B = {(5, r), (5, s), (5,t),(7,r),(7, s), (7, t), (9, r), (9, s),(9,t)}