2.1 – SETS AND SUBSETS

SET     – is a collection of numbers, distinct objects or elements.

(i)capital letters are used
(ii)every item in the bracket is called an element

i.e. the set of Natural numbers less than 8 is N = {1,2,3,4,5,7,8}

Membership:        x∈A →x is an element (member) of the set A

Notation:
• ∈ reads is “an element of”
• ∉ reads is “not an element of”
•{ } or ∅ is called an empty or null set and contains no elements.

i.e. Given: A = {1,2,3,4,5,6,7} we can say that 3 ∈ A but 9 ∉ A

Common Elements : when some items in a set are also of another set, these times are common elements and the sets are shown as overlapping circles

Subjects : lf all the elements of a set P are also elements of Q, then P is a subset of Q. lf P and Q are two sets then:

P ⊆ Q reads as “P is a subset of Q’ and means that every element in P is also an element of
Q (but not vice-versa necessarily)-

                                                   i.e. i2,3,5) ⊆ {1″,2,3,4,5,61

Universal Sets – contains all elements for that particular situation.
– All sets are subsets of the universal set.

Complement  : – the complement of set A is A’
– i.e. everything in S which is not in A

Venn Diagram :- a box containing single or overlapping circles that can be used to analyze sets in various situations.

Examples of Venn Diagrams:

For example, S= {blue, green, yellow, red} and 4 = {blue, red, green} therefore l’ is yellow.

Recall Number Sets:

N*= {1, 2,3, 4,5, 6,………………….} is the set of all counting numbers

N = {0, L,2,3, 4,5 ,6,…………………} is the set of all natural numbers

Z= {O, t 1, t 2, * 3, t4,…….. } is the set of all integers

Z* = { t, 2, 3, 4, 5, 6,…………………..} is the set of all positive integers

Z’ = {-1, -2, -3, -4, – 5,……………. i is the set of all negative integers

Q={p/q where p and q are integerand q ≠0 }isthe setof all rational numbers

6 = {real numbers} is the set of all real numbers, ie all numbers which can be placed on the number line.

Q̅ = {irrational numbers} non-terminating, non-repeating decimals.

Note:    R= {Q∪Q̅}

                                                                                  2.2- COMBINATIONS

A key idea to permutations is that order matters.

Recall: Permutations
→in an election race for president, treasurer, and secretary
→arranging letters in a word
→calling out a ticket number for a door prize

However, in some situations order doesn’t matter. We call this a combination.

For example: – in a hand of poker a “full house” is a “full house” (eg. 3 aces and 2 jacks) regard less of the order the cards were dealt.
– in Lotto Max or Lotto 6149,if your numbers match you are a winner, regardless of when the numbers were selected.

Combinations can be calculated in a similar manner as permutations. To avoid duplications, (naturally in occur in combinations) one must divide by the “extra” factorial present. This concept of “extra” factorial will be explored in the first example.

                                                                       2.3 – THE PRINCIPLE OF INCLUSION AND EXCLUSION

Mr. Wilson coaches the basketball team and the volleyball team. There are 15 members on the basketball team and 13 members on the volleyball team. Both teams are playing in a tournament at the same school and the coach orders a bus that fits 24 people.

The principle of inclusion and exclusion gives a formula for finding the number of items in the union of two or
more sets.

Two Sets:
n(A or B) = n(A)+n(B)-n(A and B) or n(A ∪ B)= n(A)+n(B)-n(A  ∩ B)

Three Sets:

                               n(A ∪ B ∪ c) = n(A) + n(B) + n(c) – n(A ∩ B)- n(A ∩ c)-n(B ∩ c) + n(A ∩ B ∩ c)