Chapter 0.3: Sets Flashcards
What is a set?
An unordered collection of unique objects.
What kind of values can a set have?
Any kind of value.
Why is set Notation helpful?
Makes writing sets less verbose
How does A = {1,2,3} read?
“A is the set containing the elements 1, 2, and 3.”
How does a ∈ {a,b,c} read? ∉?
“a is an element of the set containing the letters a, b, and c.
A = {1,b,{x, y, z}, ∅}, B = {1,b} and C = {x,y,z}. is B a subset of A? is C a subset of A?
Yes, No
What are curly braces used for?
Enclosing the elements of a set
What is ℕ?
Natural numbers set, not everyone includes 0, but the textbook does.
What does “:” read? do?
“such that”
Separates the elements in the set from the conditions to be satisfied.
How is this read: A = {x ∈ ℕ : ∃n ∈ ℕ(x = 2n) }? Can this be read more concisely?
“the set of all x in the natural numbers, such that there exists some n in the natural numbers for which x is twice n.”
“the set of all natural numbers that are even”
What are substitute symbols for ∈ that mathematicians like to use?
“|” instead of “:”
What is set builder notation?
Set building instructions that require a condition that elements must satisfy for inclusion
In set builder notation, the ____ sometimes comes before the “:”.
Condition
What is this symbol? ∅
the empty set is the set which contains no elements. Is a subset of every set
What is this symbol? 𝓤
the universe set is the set of all elements
What is this symbol? ℚ
the set of rational numbers
What is this symbol? ℤ
the set of all integers {…, -3, -2, -1, 0, 1, 2, 3, …}
What is this symbol? ℝ
the set of all real numbers. Any number that is rational or irrational
What is this symbol? ℕ
the set of all natural numbers {0, 1, 2, 3, …}
What is this symbol? P(A)
the power set of any set A that is the set of all subsets of A
Read: {1,2,3}
The set containing the elements 1, 2, and 3.
read: { x : x > 2 }
“x is the set of all x such that x is greater than 2”
What is this symbol? ⊆
“Subet of”. A ⊆ B asserts that A is a subset of B: every element of A is also an element of B
What is this symbol? ⊂
“Proper subset of”. A ⊂ B asserts that A is a proper subset of B: every element of A is also an element of B but A does not equal b
What is this symbol? ∈
“is an element of” 2 ∈ {1,2,3} is an element of the set {1,2,3}
What is this symbol? ∉
“is NOT an element of”. 4 ∉ {1,2,3} is not an element of the set {1,2,3}
What is this symbol? ⋃
Union. A ⋃ B is the union of A and B: is the set containing all elements which are elements of A or B or both.
What is this symbol? ⋂
Intersection. A ⋂ B is the intersection of A and B: the set containing all elements which are elements of both A and B
What is this symbol? \
A \ B is the set difference between A and B: the set containing all elements of A which are not elements of B.
What is this symbol? A’
(textbook has a line over A instead of an apostrophe after it) - the complement of A is the set of everything which is not an element of A.
What is this symbol? |A|
the cardinality (or size) of A is the number of elements in A.
explain ⊆ and ⊂ are analogous to ≤ and
⊆ means “is equal to or a subset of”
What is P(A)?
The set of all possible subsets of A.
What is Cardinality?
The number of elements in a set.
What is another way of explain set difference?
A ⋂ B’ the intersection of A and the complement of B.
Set operations produce ______.
Other sets.
What is a cartesian product?
A set containing the pairings of every element in one set with each element in another set.
____ are great visual representations of operations on sets?
Venn Diagrams.
Draw A ⋃ B?
Draw A ⋃ B’ or A \ B?
Draw (B ⋂ C) ⋃ (C ⋂ A’)
