Set
a collection of objects with no particular order 𝑥={1, 2, 3, A , dog, {5, 6}}
dog ∈ 𝑥 dog is an element of 𝑥
5 ∉ 𝑥 5 is not an element of 𝑥 but {5, 6} is an element of 𝑥
Function
relationship between members of one set to members of another, can be formula, description, algorithm
Empty set
∅ or { }, a set with no elements
Equal sets
Two sets are equal if they have same elements, not necessarily in the same order
Universal set
ℇ, the set containing all elements and of which all other sets are subsets
Subset
Set 𝑥 is the subset of set 𝑦 if all the elements in 𝑥 are in 𝑦, noted as 𝑥⊆𝑦
If a set has 𝑛 elements, then it has 2^𝑛 number of subsets
Proper subset
a subset 𝑦 of 𝑥 is called a proper subset of 𝑥 if 𝑦 is not equal to 𝑥
𝑦⊂𝑥 but 𝑥⊄𝑦
when 𝑦⊆𝑥, but 𝑥≠𝑦, then we say 𝑦 is properly included in 𝑥 and write 𝑦⊂𝑥
finite set
a set with a definite, or finite, number of elements
The power set of a set A (written P(A))
If a set has 𝑛 elements, then it has 2^𝑛 number of subsets
-list all possible subsets including ��
Cardinality
the number of elements in a set denoted by |𝑛|
countable infinite set
a set has infinite number of elements and the elements can be counted using 1, 2, 3...by matching them with all the natural numbers
countable - n, z, q, r
Types of numbers
natural: greater than 0 and has no decimal or fraction
Whole: natural numbers and 0
Integer: positive/negative numbers and 0
Rational: can be represented as fraction or decimal , must terminate or resolve into repeating pattern
Irrational: cannot be repre
Defining a set
example: {𝑥∈N | 𝑥<5}
| is read as such that
𝑥 is a (member) natural number such that it is less than 5
union of sets
The union of sets A and B, written A U B, is the set of elements in either A or B (or in both).
intersection of sets
The intersection of the sets of A and B, written A∩B, is the set of elements that are in both A and B.
difference of set 𝑥 in 𝑦
X/Y elements in X but not in Y
for all 𝑥, if 𝑥∈X/Y, then 𝑥∈X but 𝑥∉Y
if 𝑥∈X but 𝑥∉Y, then 𝑥∈X/Y
Y
anything outside of set Y