Set Theory
What is a Set?Real Word Examples |
In tennis, what is the meaning of a set?
Answer |
A collection of tennis games.
What other real world uses of 'sets' can you think of ?
Answer |
- sets of dishes
- set of twins
- set of forks
What is a set in mathematics?the technical definition |
In mathematics, we define a set as aof well defined, unordered, distinct
Thesecan be anything: numbers, people, animals, symbols, letters or even other sets. Sets are conventionally denoted with capital letters and elements are enclosed by curley brackets.
Examples
$
A = \{2,4,5,8, 10\}
\\
B = \{Horse, Cat, Dog, Rabbit \}
\\
C = \{ Giants, Jets, Cowboys, Eagles \}
\\
S = \{ \{1,2 \} , \{3,4,5\} , \{ A\}, \{ B\} \}
$
(Set S is a set whose elements themselves are sets. )
When are two sets equal?It's all about being'distinct' |
If you look carefully at the definitio of a set, you will notice the word distinct This is a fancy way of saying that if elements ar repeated, they still only count once.
For example, consider the following two sets, Set A and Set B:
$
A = \{ 2, 4, 6, 8 \}
\\
B = \{ 2, 2, 2, 4,4, 6, 6, 8 \}
$
Since we only care about distinct elements, set B is really just $$ \{2,4,6,8\}$$.
Therefore, A = B
Order
The definition of set states that a "set is an unordered collection of elements. What does this mean about sets C and D, below?
$
C = \{ 10, 27, 4 \}
\\
D = \{ 4,10,27 \}
$
Answer |
Practice Identify Equal Sets
Directions : Look at each pair of sets below and determine if they are equal
1) | A= {-1,0,1,2 } and B= {-1,0,1 } |
Equal
Not Equal |
2) | C = {10, 20, 30, 40} and D = {40, 30,20,10} |
Equal
Not Equal |
3) | E= { red, red, blue, green, blue} and F= {red,green,blue} |
Equal
Not Equal |
4) | G = {a, e, i, o, u} and H = { e,e,a,o,u } |
Equal
Not Equal |
5) | I = { Queen, Pawn, Knight , Bishop} and J = { Bishop, Pawn, Queen, Rook} |
Equal
Not Equal |
6) | I = { ♥ , ♠, ♦, ∇ , ♣ } and J = { ♥ , ♠, ♦, • , ♣ } |
Equal
Not Equal |
Elements of a set
Given the following two sets:
$ J = \{2, 4, 5, 10, 14 \} \\ K = \{ \pi , \sqrt{3} , -2\pi, - \sqrt{5} , \frac{2\pi}{3} \} $
If we want to say that "2 is an element in Set J", but that "$\pi$ is not an element of Set J", we have a simplie mathmeathical notation to concisely write this:
$ J = \{2, 4, 5, 10, 14 \} \\ K = \{ \pi , \sqrt{3} , -2\pi, - \sqrt{5} , \frac{2\pi}{3} \} $
Set Notation | Meaning |
$2{\in} J $ | "2 is an element in set J" |
$2 {\notin} J $ | "2 is not an element in set J" |
Practice With Set Notation
Directions :The questions below refer to sets J and K above. Choose the correct symbol that goes in the blank.
1) 5
2) 2$\pi$ k
3) 12 J
4) $\sqrt{3}$ K
5) $\sqrt{5}$
6) 6 J
J
2) 2$\pi$ k
3) 12 J
4) $\sqrt{3}$ K
5) $\sqrt{5}$
6) 6 J
Cardinatlity, Null and more