Sets - Venn Diagram

Sets - Venn Diagram

Set

A set is a well-defined collection or grouping of objects.

Examples:

1. The group “colors in the Philippine Flag” is well-defined. We can easily identify all the colors in the Philippine Flag

2. The group “colors of flowers” is not well-defined. There are so many different varieties of flowers that it will be very difficult to identify all their colors.

Since no. 1 is well defined, we can describe it as “a set of colors in the Philippine Flag”.

No. 2 is not well-defined, we cannot call it a set.

Multiple Select

Which of the following are well defined groups?

1. factors of 21

• well-defined

• not well-defined

Multiple Select

Which of the following are well defined groups?

expensive school bags

• well-defined

• not well-defined

Multiple Select

Which of the following are well defined groups?

distinct letters in the word “Welcome”

• well-defined

• not well-defined

Multiple Select

Which of the following are well defined groups?

good looking mentors of AMSLI

• well-defined

• not well-defined

Multiple Select

Which of the following are well defined groups?

diligent students of AMEP

• well-defined

• not well-defined

Multiple Select

Which of the following are well defined groups?

odd numbers less than 90 but greater than 60

• well-defined

• not well-defined

Multiple Select

Which of the following are well defined groups?

a movie

• well-defined

• not well-defined

Multiple Select

Which of the following are well defined groups?

a form of transportation

• well-defined

• not well-defined

Multiple Select

Which of the following are well defined groups?

a continent

• well-defined

• not well-defined

Multiple Select

Which of the following are well defined groups?

easy questions in mathematics test.

• well-defined

• not well-defined

Elements of a Set

The objects included in the set are called elements or members of the set.

The symbol, ∈ reads “is an element of “, while ∉ reads “is not an element of”.

Ways to write a set

• 1. descriptive method - the elements of the set are described within a specific category

  Example: Set A is a set of counting numbers

• 2. roster form - the elements are enumerated and enclosed by braces

  Example: A = {1, 2, 3, 4, 5, ... }

• 3. set-builder notation - the specific description refers to an element of the set

  Example: A = {x|x is a counting number}

  [Read as “x such that x is a counting number”]

Write each set in descriptive method.

• 1. A = {September}

• 2. B = {t, r, a, i, n}

• 3. C = a, e, i, o, u

• 4. D = {thumb, index, middle, ring, pinky}

• 5. E = {0, 1, 2,... , 9}

• 6. P = {square, rectangle, parallelogram, rhombus, trapezoid}

Write the described sets in roster form.

• 1. A is the set of planets nearer the Sun and before planet Earth.

• 2. B is the set of multiples of 11 greater than 30 but less than 100

  Ans. B = {33, 44, 55, 66, 77, 88, 99}

• 3. C is the set of the last five letters in the alphabet.

• 4. D is a set of positive odd number less than 12.

• 5. E = {fractions whose denominator is seven and is between 1 and 2}

Write the described sets in set builder notation.

• 1. F is a set of cities in Metro Manila.

• 2. G is the set of provinces in Luzon

• 3. H is the set of natural numbers less than 50 that are perfect squares.

• 4. I is the set of positive numbers less than 100 that are perfect cubes

• 5. J is a set of male AMSLI mentors

Cardinality of a Set

The cardinality of a set tells how many elements are in the set. The symbol n(B) reads “the number of elements of set B” or “the cardinality of set B”.

Example. Given: B = { whole numbers between 5 and 12}

The elements of set B are the whole numbers 6, 7, 8, 9, 10, 11 and 12.

There are 7 elements of set B. The cardinality of set B is 7. In symbol, n(B) = 7.

Classification of Sets

• 1. null or empty set has no element. Empty sets can be represented by  or  .

• 2. unit set has only one element.

• 3. finite set has a cardinal number of elements

• 4. infinite set - it is not possible to end the counting of the elements of the set

Subsets

The elements of a set can be subdivided into different sets which are called subsets. The specified set from which subsets are derived is called the universal set.

Example. Let U = {the first three letters of the English alphabet}.

In roster form, U = {a, b, c}

The subsets of U are : A = {a }. ; B = { b } ; C = { c}; D = {a, b } ; E = { b, c} ; F = {a, c}; G = {a, b, c}

We represent “A is a subset of U” by the symbol A  U.

Since set U contained all the elements of set A or set U may have more than the elements

contained in set A, we say, “U is the superset of A”. In symbol, we write U  A if “ A is a proper

superset of U” and U  A if “A is a superset of U”.

Proper Subset

A proper subset is when all the elements of that subset belong to set U but at least there is one element of U not found in that subset. In symbol, A U means “A is a proper subset of U”

Improper Subset

An improper subset is when all the elements of that subset belongs to set U. This really

means that the improper subset is equal to the universal set. In symbol, D  U.

Every set is a subset of itself. An empty set is a subset of any set. Any given set has 2^n number

of subsets where n is equal to the number of elements, or n is the cardinality of the specified

set. We say that 2^n is the power set of the given set.

Multiple Select

Given: M is a set of all the months in a year.

R is a set of whole numbers less than 30.

Tell whether each statement is true or false.

1. Sunday  M

(Sunday is an element of M)

• True

• False

Multiple Select

Given: M is a set of all the months in a year.

R is a set of whole numbers less than 30.

Tell whether each statement is true or false.

2. 15  R

(15 is an element of R)

• True

• False

Multiple Select

Given: M is a set of all the months in a year.

R is a set of whole numbers less than 30.

Tell whether each statement is true or false.

3. June  M

(June is not an element of M)

• True

• False

Multiple Select

Given: M is a set of all the months in a year.

R is a set of whole numbers less than 30.

Tell whether each statement is true or false.

4. 21  R

(21 is an element of R)

• True

• False

Multiple Select

Given: M is a set of all the months in a year.

R is a set of whole numbers less than 30.

Tell whether each statement is true or false.

5. 11  R

(11 is not an element of R)

• True

• False

Multiple Select

Given: M is a set of all the months in a year.

R is a set of whole numbers less than 30.

Tell whether each statement is true or false.

6. {2, 3, 5, 7, 11,13, 17, 19, 21, 23} is an improper subset of R

6. {2, 3, 5, 7, 11,13, 17, 19, 21, 23} R

• True

• False

Multiple Select

Given: M is a set of all the months in a year.

R is a set of whole numbers less than 30.

Tell whether each statement is true or false.

7. { }  M

{ } is a subset of M

• True

• False

Multiple Select

Given: M is a set of all the months in a year.

R is a set of whole numbers less than 30.

Tell whether each statement is true or false.

8. {7, 13, 29}  R

{7, 13, 29} is a subset of R

• True

• False

Multiple Select

Given: M is a set of all the months in a year.

R is a set of whole numbers less than 30.

Tell whether each statement is true or false.

9. {Friday, January, Monday}  M

{Friday, January, Monday} is not a subest of M

• True

• False

Multiple Select

Given: M is a set of all the months in a year.

R is a set of whole numbers less than 30.

Tell whether each statement is true or false.

10. Set R is a subset of the set of integers.

• True

• False

Given: Let P = {all prime numbers between 5 and 15}

• 1. List all the elements of set P.

• 2. List all the subsets of P that contains one element.

• 3. List all the subsets of P that contains two elements.

• 4. Identify the improper subset of set P.

• 5. How many subsets do set P have?

Set Relations

Set A is equal or identical to set B, A = B, if they have the same elements.

In equivalent sets, the elements of the two sets can be paired exactly or they can be put

into a one-to-one correspondence.

Joint and Disjoint Sets

Sets which have common elements are called joint sets.



Sets which have no common elements are called disjoint sets.

Multiple Choice

Determine if the pairs of sets are equivalent, equal, both or neither.

1. A = {x|x is a day of the week}

B = {1, 3, 5, 7, 9, 11, 13}

• Equivalent

• Equal

• Both

• Neither

Multiple Choice

Determine if the pairs of sets are equivalent, equal, both or neither.

2. C = {consonants in the word “hymn”}

D = {letters in the word “nymph”}

• Equivalent

• Equal

• Both

• Neither

Multiple Choice

Determine if the pairs of sets are equivalent, equal, both or neither.

3. E = {all people with three hands}.

F = {all circles with two vertices}

• Equivalent

• Equal

• Both

• Neither

Multiple Choice

Determine if the pairs of sets are equivalent, equal, both or neither.

4. G = {red, orange, yellow, green, blue, indigo, violet}.

H = {x|x is a color of the

rainbow}

• Equivalent

• Equal

• Both

• Neither

Multiple Choice

Determine if the pairs of sets are equivalent, equal, both or neither.

5. J = {x|x is a vowel in the word “system”}

K = {x|x is a vowel in the word “psalms}

• Equivalent

• Equal

• Both

• Neither

Multiple Select

Determine if the pairs of sets are joint or disjoint sets.

1. V = {x|x is a subject in Mathematics}

Z = {English, Filipino, Science}

• Joint

• Disjoint

Multiple Select

Determine if the pairs of sets are joint or disjoint sets.

2. N = {proper fractions whose denominator is 15}

P = {proper fractions whose denominator is 3}

• Joint

• Disjoint

Multiple Select

Determine if the pairs of sets are joint or disjoint sets.

3. Q = {x|x is a rectangle with equal sides}

R = {x|x is a triangle with equal sides}

• Joint

• Disjoint

Multiple Select

Determine if the pairs of sets are joint or disjoint sets.

4. S = {x|x is a whole number greater than 15 but less than 17}

T = {x|x is a multiple of 4 and x is less than 18}

• Joint

• Disjoint

Multiple Select

Determine if the pairs of sets are joint or disjoint sets.

5. L = {even whole numbers between 31 and 39}

M = {prime numbers between 31 and 39}

• Joint

• Disjoint

Multiple Select

Given: M is a set of all the months in a year.

R is a set of whole numbers less than 30.

Tell whether each statement is true or false.

4. 21  R

(21 is an element of R)

• True

• False

Multiple Select

Given: M is a set of all the months in a year.

R is a set of whole numbers less than 30.

Tell whether each statement is true or false.

5. 11  R

(11 is not an element of R)

• True

• False