Set Operations

1. With any two sets “A” and “B” there is associated a third set “C” satisfying the property that C = { X/X Є AV x Є B }

In words: “C” is equal to X, such that X is belong to “A” or X is an element of B

“C” is called the union of “A” and “B” we denote the set C symbolically as C = A U B

Example:
A = {3, 4, 5, 6, 7}
B = {2, 4, 6, 8, 10}
A U B = {2, 3, 4, 5, 6, 7, 8, 10}

2. With any two sets A & B there is associated A third set “D” satisfying the property that D = { X/X Є A ۸ X Є B}

In words: D equals X such that X is an element of set “A” and x is a member of B.

“D” is called the intersection of sets A and B, and we denote the set D symbolically as D = A B

3. With any two sets A and B there is associated A third set “C” satisfying the property that C = { X/X Є A ۸ X € B}. We denote the set symbolically as C = A – B, and call C the relative complement or difference of A and B.

Example:
A = { a, b, c, d, e, f}
B = { a, e, i, o, u}
A – B = {b, c, d, f} and B – A = {i, o, u}

4. If A is a subset of U, then the set of an elements contained in U that are not elements of A is called the complement of A in U and is designated by Ă then Ă = {X/X Є U ۸ X € A}

Example: Consider the universal set of an counting nos. and the set A of counting numbers less than 100 then

U = {1, 2, 3, 4, ……} A = { 1, 2, 3,……99}

Ă = {100, 101, 103…….}

5. The set product or cartesian product of two sets A and B is the set of an possible ordered pairs (a, b) where a is in A and b is in B. We symbolize this set of ordered pairs by A X B and write,
A X B = {(a,b) / a Є A ۸ b Є B }

Example:

If A = {1, 2} and B = {x, y} then A X B = { (1, x), (1, y), (2, x) , (2, y}} and B X A = {(X, 1), (X, 2), (Y, 1), (Y, 2)}

Kinds of set

1. Finite set – countable
Example: Sets A, B, C, D are finite sets

2. Infinite set – uncountable
Example: Set E is an infinite set

3. Empty or null set – has no element
Example: A = { }

4. Equal set – set A and set B are equal set if the elements of set A is exactly the element of set B.
Example:
A = {set of an even counting number of one digit} = {2,4,6,8}
B = {set of an integral multiples of two having one digit = {2,4,6,8}

5. Equivalent set – two sets are equivalent if there exists a one-to-one correspondence between elements of the two sets.
Example:
A = {1, 2, 3, 4,5} - x coordinate
B = {6, 7, 8, 9, 10} – y coordinate

then “A” is equivalent to B. We can construct the relation of set A and set B.

{ (1,6}, (2,7), (3,8), (4,4), (5,10) }

6. Subset – set whose elements are members of the given set A = {1,2,3,4,5,8}, B = {2,4,8}

7. Universal Set – totality of the given set with consideration. The set from which we select elements to form A given set is called universal.
Example:
Set A = {1, 2, 3, 4, 5, 8} is a universal set
Set B = {2, 4, 8} is a subset of set A

8. Disjoint Set – sets that has no common element ; if two sets have no element in common, the sets are called disjoint sets.

Methods of Writing Set

Example: Roster Method and Rule Method

Methods of Writing Set

1. Roster or tabular method

The elements of the set are enumerated and separated by comma.

2. Rule method or set builder

A, descriptive phrase is used to describe the elements of the set

Sets Definition and Examples

Set

Definition:
Set is a well-defined collection of things or objects

Note:
Sets maybe denoted by capital letters such as A,B,X, Y

An element or member of a set is a thing that belongs to the set and maybe denoted by small letters such as a,b,c……..x,y.

The members of the set are enclose in braces { }, with a comma separating the members.

Example:

The set “A” whose members are ETHEL, CYNTHIA, CHELO, we usually use the symbol.

A = {ETHEL, CYNTHIA, CHELO}

ETHEL Є A
- Read as ETHEL is an element of set A
- Read as ETHEL belongs to set A
- Read as ETHEL is a member of set A

Empty Set and Set

- A set is a collection of things
- An element or member of a set is a thing that belongs to the set.

* There are many words which we use in everyday language that have the same meaning as the word set.

Example:

1. A herb of cattle is a set of cattle
2. A flock of birds is a set of birds
3. A squadron of planes is a set of planes
4. a school of fish is a set of fish
5. A regiment of soldiers is a set of soldiers

* The members of the set are enclosed in braces, { }, with a comma separation the members and to identify sets we often name them by capital letters.

Example:

1. The Set “A” whose members are Ethel, Emerson and Merecel. We usually use the symbol

A = {Ethel, Emerson, Merecel}

2. The Set “B” of days of the week

B = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}

The set C of words to distinguish two faces of a coin

C = {Heads, Tails}

* The set that has no elements is called the empty set, we use the symbol Ǿ to indicate the empty set.

Example of Empty set:

1. the set of whole numbers by 9 and 10.
2. the set of four-sided triangles.
3. the set of astronauts who have landed on the planet Pluto
4. the set of icebergs in the sahara desert
5. the set of people with two heads
6. the set of pink elephants
7. the set of purple cows

Permutation Formula and Example

Permutation

Each different arrangement or ordered set of object is caused a permutation of those objects.

- if A = {a1, a2, a3……. An} is any set of n elements then any arrangement of the elements of “A” along a line is called a permutation of the elements of A.


All the permutation of the elements of the set is given by the formula:

P = n! where n = no. of elements

Problem:

How many permutations can be made from the word PINOY”

Solution:

PINOY – consist of 5 letters

P = 5! = 120 permutations

The total no. of permutations of n objects taken r at a time, P(n,r) is given by the expression.

P (n,r) = nPr = n!/(n-r)!

Problem:

Find the no. of permutations of the four integers 1,2,3,4 taken two at a time.

Solution:

n = 4, r = 2

4P2 = n!/(n-r)! = 4!/(4-2)! = 4!/2! = 4.3.2.1 / 2.1 = 12

Statistics Probability Sample Problems

1. At a certain canteen, Doris can choose merienda from three drinks (Coke, Pepsi, Gulaman) and four sandwiches from (bacon, chicken, tuna, egg). In how many ways.

Solution:

D = {Coke, Pepsi, Gulaman}
N(D) = 3
S = {Bacon, Chicken, Tuna, Egg}
N(S) = 4

N1 . N2 = 3 x 4 = 12 ways

2. Two dice are rowed, in how many ways can they fall? If 3 dice are rowed? and if 4 dice are rowed?

For two dice
N1 = 6
N2 = 6
N1.N2 = 6 x 6 = 36 ways

For three dice
N1 . N2 . N3
6 x 6 x 6 = 216 ways

For four dice
N1.N2.N3.N4
6 x 6 x 6 x 6 = 296 ways

3. Using the digits 1,2,3,4,5,6, How many two-digit can be formed if a) repetition is allowed b) repetition is not allowed. How many numbers do we have to choose from the given set, they are 6 numbers.

Solution:

a) Repetition is allowed
6 x 6 = 36 ways

b) Repetition is not allowed
6 x 5 = 30 ways