⇰SETS is a collection of objects that have something in common or follow rule. the objects u the set are called its elements. Set notation uses curly braces, with elements separated by commas.
| EXAMPLE⬎ |
 |
| A = {coat, hat, scarf, gloves, boots}, where A is the name of the set, and the braces indicate that the objects written between them belong to the set. |
Numerical Sets
When we define a set, all we have to specify is a common characteristic.
Set of even numbers: {..., -4, -2, 0, 2, 4, ...}
Set of odd numbers: {..., -3, -1, 1, 3, ...}
Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}
Positive multiples of 3 that are less than 10: {3, 6, 9}
And the list goes on. We can come up with all different types of sets.
There can also be sets of numbers that have no common property, they are just defined that way. For example:
{2, 3, 6, 828, 3839, 8827}
{4, 5, 6, 10, 21}
{2, 949, 48282, 42882959, 119484203}
Every object in a set is unique: The same object cannot be included in the set more than once.
Look at some more examples of sets.
⇝ What is the set of all fingers?
SOLUTION : P = {thumb, index, middle, ring, little}
⤐What is the set of all even whole numbers between 0 and 100?
SOLUTION : Q = {2, 4, 6, 8}
⚠Note that the use of the word between means that the range of numbers given is not inclusive. As a result, the numbers 0 and 10 are not listed as elements in this set⚠
Empty Set or Null Set:
A set which does not contain any element is called an empty set, or the null set or the void set and it is denoted by ∅ and is read as phi. In roster form, ∅ is denoted by {}. An empty set is a finite set, since the number of elements in an empty set is finite, i.e., 0.
FOR EXAMPLE :
(a) The set of whole numbers less than 0.
(b) Clearly there is no whole number less than 0.
Therefore, it is an empty set.
(c) N = {x : x ∈ N, 3 < x < 4}
• Let A = {x : 2 < x < 3, x is a natural number}
Here A is an empty set because there is no natural number between
2 and 3.
• Let B = {x : x is a composite number less than 4}.
Here B is an empty set because there is no composite number less than 4.
Note🔔
∅ ≠ {0} ∴ has no element.
{0} is a set which has one element 0.
The cardinal number of an empty set, i.e., n(∅) = 0
😶 TRY OUT THIS QUESTIONS
1. If U = {1, 3, 5, 7, 9, 11, 13}, then which of the following are subsets of U.
B = {2, 4}
A = {0}
C = {1, 9, 5, 13}
D = {5, 11, 1}
E = {13, 7, 9, 11, 5, 3, 1}
F = {2, 3, 4, 5}
2. Let A = {2, 3, 4, 5, 6, 7} B = {2, 4, 7, 8) C = {2, 4}. Fill in the blanks by ⊂ or ⊄ to make the resulting statements true.
(a) B __ A
(b) C __ A
(c) B __ C
(d) ∅ __ B
(e) C __ C
(f) C __ B
This is just multiplication and division. The “!” is the factorial symbol. That’s just a special way of multiplying numbers. To get a factorial, multiply the number by each number below it until you get to 1. For example:
4! = 4 x 3 x 2 x 1 = 24
2! = 2 x 1 = 2
Permutations and Combinations: Sample Problems
Sample problem #1: Five bingo numbers are being picked from a ball containing 100 bingo numbers. How many possible ways are there for picking different numbers?
Step 1: Figure out if you have permutations or combinations. Order doesn’t matter in Bingo. Or for that matter, most lottery games. As order doesn’t matter, it’s a combination.
Step 2: Put your numbers into the formula. The number of items (Bingo numbers) is “n.” And “k” is the number of items you want to put in order. You have 100 Bingo numbers and are picking 5 at a time, so:
Step 3: Solve
Step 3: Solve
![[IMAGE]](http://www.mathgoodies.com/lessons/vol6/images/marbles_jar.gif)
