Set Theory Flashcards

Covers the basics of set theory

1
Q

what is the set B (stylised)?

A

binary numbers - {0,1}

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2
Q

why can sets with elements listed in a different order/with duplicates be considered equal?

A

a set is defined solely in terms of its members

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3
Q

A = B iff…

A

A ⊆ B and B ⊆ A

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4
Q

¬(A⊆B)

A

A ⊈ B

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5
Q

the universe of discourse consists of

A

all elements under consideration

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5
Q

what are the 3 properties of the subset relation?

A

reflexive, antisymmetric and transitive

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6
Q

describe how the subset relation is reflexive

A

A ⊆ A for all sets A

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7
Q

describe how the set relation is anti-symmetric

A

if a and b are related in both directions, they must be equal. e.g iff A ⊆ B and B ⊆ A, A = B

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8
Q

describe how the subset relation is transitive

A

if A ⊆ B and B ⊆ C, A ⊆ C

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9
Q

what is the least set with respect to inclusion?

A

the empty set since it is contained in any other set

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10
Q

how to use set builder notation to define B, the subset of A

A

B = { x ∈ A : x has property P }

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11
Q

write { Dan”, Felix } using set builder notation

A

{ x : x = Dan or x = Felix }

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12
Q

what is Russell’s paradox

A

the set of sets that do not contain themselves CANNOT exist in naive set theory

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13
Q

the importance of Russell’s paradox in set theory

A

it should not be possible to construct sets of all sets

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14
Q

two sets are disjoint if…

A

they have no elements in common

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15
Q

the intersection of two disjoint sets

A

the empty set

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16
Q

|A|

A

the cardinality of A

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17
Q

what is the inclusion-exclusion principle

A

|A U B| = |A| + |B| - |A ∩ B|

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18
Q

A \ B in set builder notation

A

{ x ∈ A : x ∉ B }

18
Q

x ∈ { A \ B } iff

A

x ∈ A and x ∉ B

18
Q

construct the set A’ using only the universe discourse and the set A

19
Q

P(A)

A

the set containing all subsets of A (including the empty set and itself)

20
Q

a set of 4 elements has how many subsets

20
Q

what trick can you use to figure out the number of elements in a powerset?

A

Pascal’s Triangle

20
what is a family of sets F?
a collection of sets (where every element in F is a set)
21
what is the union of the family of sets F?
{ x : x ∈ A for some A ∈ F }
22
what is the intersection of a family of sets?
{ x : x ∈ A for all A ∈ F }
23
both the union and intersection of a family of sets are..
sets of elements (arbritrary x's)
24
Alternative way to write U F if F = {A, B, C }
U{A, B, C}
24
if F = {A,B,C}, what is U F shorthand for?
A U B U C
25
U{A, B} in written english
the union of all sets within the family {A,B}
26
what is an ordered pair?
a pair of objects with a first coordinate and a second coordinate
27
(a,b) ∈ A ✕ B iff
a ∈ A and b ∈ B
28
two ways to denote the cartesian product across all real numbers
R ✕ R or R^2
29
R ✕ R denotes the set of..
points in the xy plane
30
key value pairs are always ordered pairs from..
the Cartesian product Keys ✕ Values
31
A^n corresponds to
the cartesian product of n copies of the set A (e.g an R vector over 4)
32
the number is elements in a product is
the product of each individual set's cardinality
33
|A ✕ {}| | A ✕ {} |
0
34
A ✕ {}
{}
35
X ⊆ Y Hence X\Y =
{}
36
X ⊆ Y Hence X' U Y =
The universal set
37
from the identity (P ⇒ Q) ⇔ (¬P ∨ Q), we get the identity P ⊆ Q = ...
P' U Q = universe set
38
property of sets that corresponds to the contrapositive law
A ⊆ B iff A' ⊆ B'
39