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Sets - Coggle Diagram
Sets
Sets & Subsets
Set structure/rules
Rules
The order of a set Is unimportant
Repeating an element does not change the set
Structure
A set is a collection of varied or identical objects called elements. Sets can be expressed in Roster notation with each element being enclosed by curly braces, and separated by commas
Set Notation
An element in a set is denoted as a ∈ A; if an element is not in a set, it is denoted as a ∉ A. Capital letters are used to denote set names and a stand for variables. Variables can include more than one element.
The empty set is a set with no elements, meaning a ∉ ∅ is a true statement. It is also known as the null set, and is denoted with {}.
Cardinality: The number of distinct elements in a set; denoted with |A|
Finite Sets
A finite set is a set that contains zero elements, or that can be numbered 1 to n
Infinite Sets
An infinite set is one that is not finite
Ellipses are used in longer sets to denote missing in between elements when the pattern between each element Is known
Subsets
A is a subset of B if every element of A is an element of B; denoted as A ⊆ B; in this case, A = B when both are subsets of each other. A proper subset is when one is a subset of another, but not vice versa(A ⊆ B but B ⊈ A), denoted as A ⊂ B
Applications of Sets
Common Mathematical sets
Z: set of all integers(negative numbers, zero and positive numbers
Q: set of all rational numbers(all numbers that can be expressed as a/b where b ≠ 0
N: set of Natural numbers; all numbers greater than or equal to zero
R: set of all real numbers
Universal Sets & Venn Diagrams
Universal sets are all the elements in a particular context(ie. students in a school)
Each set in a universal set is a circle in a Venn diagram. The elements that aren't elements of a set are outside the circle, but still inside the universal set
Power Sets
Elements of sets can also be sets themselves(ie. {{1,2}, {3,5}, {8, 10}} has three elements, however 1 is not an element of A
The set containing the empty set is not the same as the empty set
P(A) is all the possible subsets of A; ie. A = {1,2,3} so P(A) = {ø, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {3,3}, {1,2,3}}
Set Identities
Set Operations
Cartesian Products
Partitions