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

Statistics Probability: Definitions, Principles and Samples

Probability which connotes the “chance” or the “likelihood” that something will happen or occur is an interesting and fascinating area of mathematics.

Probability – the part of mathematics that deals with the questions “how likely” is called probability or the theory of probability.

Probability – is a measure of certainty, its scale is from 0 to 1. A probability of zero indicates that there is no chance at all that an event will happen or occur. A probability of one (1) indicates absolute certainty that an event will happen. Absolute certainly rarely happens in lifes.

1. Experiment

Activity that can be done repeatedly.

Examples:
1. Tossing a coin
2. Rolling a pie


2. Sample Space – set of all possible outcomes in an experiment(s)

Examples:
a.) S = {H,T} n(S) = 2
b.) S = {1,2,3,4,5,6} n(S_ = 6
c.) S = {Rod, Ed, Emer} n(S) = 3

3. Sample Point – an element in the sample space

Examples
a.) H is a sample point
T is a sample point

4. Event – is a subset of sample space

Example:
Getting an even number when you roll a die is an event

S = {1,2,3,4,5,6}
E = {2,4,6}
n (E) = 3

Counting Techniques

N1 . N2 . N3 . N4 …..Nn (where N = event)

Fundamental Principles

If one thing can be done independently in N1 different ways and if a second thing can be done independently in N2 different ways and so on. Then the total number of ways in which all the things may be done in the stated order is N1 . N2 . N3 . N4 ……….