Set Theory Flashcards
x ∈ A
x is an element of A
A = {x1, x2, …, xn}
A is a set containing x1, x2, …, xn
A = {x: P(x)}
A is the set of elements x satisfying property P
A ⊆ B
A is a subset of B
A ⊂ B
A is a proper subset of B (A is a subset of B and A does not equal B)
Ø
null set or empty set
φ(A)
the powerset of A (the set of all subsets of A)
If A = {1,2}, what is the powerset of A?
φ(A) = {Ø, {1}, {2}, {1, 2}}
If |A| = n, then what is |φ(A)|?
|φ(A)| = 2^n
How do you prove A ⊆ B?
show that for every x ∈ A, that x ∈ B
How do you prove A = B?
show that A ⊆ B and B ⊆ A
What is the size of a set called?
the cardinality of a set
∪
union
∩
intersection
A^c
complement, everything not contained in A
what is the transitive property (of sets?)
if A ⊆ B and B ⊆ C, then A ⊆ C
A ∩ ( B U C) is equivalent to…
(A ∩ B) U (A ∩ C)
disjoint
A and B are disjoint if A ∩ B = Ø
cross product
A x B = {(a, b): a ∈ A and b ∈ B}
if A = {1, 2} and B = {3, 4}, what is A x B?
A x B = {(1, 3), (1, 4), (2, 3), (2, 4)}
if |A| = n and |B| = m, what is |A x B|?
|A x B| = nm
note: how to read family of sets, union
union of A as A varies over the collection of sets
note: how to read family of sets, intersection
intersection of A as A varies over the the collection of sets
what is Δ?
indexing a set
A = {Bα: α ∈ Δ}
a set of sets, B1, B2, ….
pairwise disjoint
in a collection of sets, any pair of sets has an empty intersection
what do de morgan’s laws concern?
the compliments of sets and unions and intersections of sets
