Set Theory Lecture 1 Flashcards
What is a set?
An unordered collection of elements.
What is an example of a set?
A := {1,2,3} or DaysOfWeek := {Mon,Tue,Wed,Thu,Fri,Sat,Sun}.
What is the notation for natural numbers (N)?
N := {0,1,2,3,…}.
What is the notation for integer numbers (Z)?
Z := {…,-2,-1,0,1,2,…}.
What does a ∈ A mean?
a is an element of set A.
What does a ∉ A mean?
a is not an element of set A.
What does A ⊆ B mean?
A is a subset of B, meaning all elements of A are in B.
What does A ⊈ B mean?
At least one element of A is not in B.
What is the definition of set equality?
A = B if and only if A ⊆ B and B ⊆ A.
What is the empty set?
A set with no elements, denoted as ∅ or {}.
What is the number of elements in {4,3,3,2,1,2}?
4 (since sets do not count duplicates).
What is the universe (U) in set theory?
The context in which all sets exist.
What is the complement of a set A?
A′ = {x ∈ U | x ∉ A}.
What is the union of sets A and B?
A ∪ B = {x ∈ U | x ∈ A or x ∈ B}.
What is the intersection of sets A and B?
A ∩ B = {x ∈ U | x ∈ A and x ∈ B}.
What is A \ B (set difference)?
The elements in A that are not in B.
What is the symmetric difference of A and B?
A △ B = (A ∪ B) ∩ (A ∩ B)′.
What are the three properties of a partition?
- Parts are non-empty, 2. Parts do not overlap, 3. Parts cover the entire set.
What is the rule of sum for partitions?
A = #A1 + #A2 + … + #An.
What does the Venn diagram illustrate?
The relationships between sets visually.
What is De Morgan’s Law for complements?
(A ∩ B)′ = A′ ∪ B′ and (A ∪ B)′ = A′ ∩ B′.
What is the formula for A ∩ (B \ C)?
A ∩ (B \ C) = (A ∩ B) \ C.
What does #(A ∪ B)′ mean?
The number of elements in the universe that are in neither A nor B.
What is the commutative property of sets?
A ∩ B = B ∩ A and A ∪ B = B ∪ A.
What is the associative property of sets?
A ∩ (B ∩ C) = (A ∩ B) ∩ C and A ∪ (B ∪ C) = (A ∪ B) ∪ C.
What is the idempotence property of sets?
A ∩ A = A and A ∪ A = A.
What is the involution property of sets?
(A′)′ = A.
What is the distributive law of sets?
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).
What is a tautology in set algebra?
A statement that is always true, like A ∪ U = U.
What is an example of a set equation verification?
(A′ ∩ B)′ ∪ (C ∩ A′)′ = A ∪ (B ∩ C)′.
