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105 Cards in this Set
- Front
- Back
Set Identities
A⋃∅ A⋂U |
Identity Laws
A A |
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Set Identities
A⋃U A⋂∅ |
Domination Laws
U ∅ |
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Set Identities
A⋃A A⋂A |
Idempotent Laws
A A |
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Set Identities
A⋃B A⋂B |
Commutative Laws
B⋃A B⋂A |
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Set Identities
A⋃(B⋃C) A⋂(B⋂C) |
Associative Laws
(A⋃B)⋃C (A⋂B)⋂C |
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Set Identities
A⋂(B⋃C) A⋃(B⋂C) |
Distributive Laws
(A⋂B)⋃(A⋂C) (A⋃B)⋂(A⋃C) |
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Set Identities
(A⋂B)⋃(A⋂C) (A⋃B)⋂(A⋃C) |
Reverse Distributive Laws
A⋂(B⋃C) A⋃(B⋂C) |
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Set Identities
~(A⋃B) ~(A⋂B) Those should be lines over the whole top |
De Morgan's Laws
~A⋂~B ~A⋃~B Those should be lines over individual letters |
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Set Identities
~A⋂~B ~A⋃~B Those should be lines over individual letters |
Reverse De Morgan's Laws
~(A⋃B) ~(A⋂B) Those should be lines over the whole top |
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Set Identities
A⋃(A⋂B) A⋂(A⋃B) |
Absorption Laws
A A |
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Set Identities
A⋃~A A⋂~A Should be lines on top |
Complement Laws
U ∅ |
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Truth sets
{x|P(x)} explain the parts |
The elements of the set for which the values of x make the second part true - the elementhood test
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y∈{x|P(x)}
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P(y)
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y∉{x|P(x)}
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~P(y)
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A is a truthset for the proposition P(x)
What is A in set form What is meant by y∈A y∉A |
P(y)
~P(y) |
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Redefine
{x∈U|P(x)} |
{x|x∈U^P(x)}
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Redefine
{x|x∈U^P(x)} |
{x∈U|P(x)}
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Redefine
y∈{x∈U|P(x)} |
y∈U^P(y)
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Redefine: What is it?
A⋂B |
It is a set
{x|x∈A^x∈B} If A and B have truth tests, then it could mean P(x)^Q(x) |
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What is it?
x ∈ A⋂B |
It is a statement
x∈A^x∈B and if they had propositions P(x)^Q(x) |
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Redefine: What is it?
A⋃B |
It is a set
{x|x∈A v x∈B} If A and B have truth tests, then it could mean P(x) v Q(x) |
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What is it?
x ∈ A⋃B |
It is a statement
x∈A v x∈B and if they had propositions P(x) v Q(x) |
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Redefine: What is it?
A/B |
It is a set
{x|x∈A ^ x∉B} If A and B have truth tests, then it could mean P(x) ^ ~Q(x) |
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What is it?
x ∈ A/B |
It is a statement
x∈A ^ x∉B and if they had propositions P(x) ^ ~Q(x) |
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Redefine:
{x|x∈A^x∈B} |
A⋂B
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Redefine:
x∈A^x∈B |
x ∈ A⋂B
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Redefine:
{x|x∈A v x∈B} |
A⋃B
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x∈A v x∈B
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x ∈ A⋃B
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Redefine:
{x|x∈A ^ x∉B} |
A/B
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Redefine:
x∈A ^ x∉B |
x ∈ A/B
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What is it? Name it
Redefine: A∆B (2 definitions) |
It's a set
Symmetric Difference (A\B)⋃(B\A) (A⋃B)\(A⋂B) |
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Redefine: what is it?
A⊆B (2 definitions) |
It is a statement:
∀x(x∈A → x∈B) ∀x∈A(x∈B) |
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Redefine: What is it?
A=B (2 definitions) |
It is a Statement
∀x(x∈A ↔ x∈B) A⊆B ^ B⊆A |
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Redefine:
(A\B)⋃(B\A) (A⋃B)\(A⋂B) |
A∆B
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Redefine:
∀x(x∈A → x∈B) ∀x∈A(x∈B) |
A⊆B
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Redefine:
∀x(x∈A ↔ x∈B) A⊆B ^ B⊆A |
A=B
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What is it? What is it called?
Redefine A⋂B=∅ |
A statement
Called disjoint Allx, x is part of the first part iff x is part of the second part? or This first part is a subset of the second, and the second is a subset of the first? |
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What is it?
Re-express: {pi|i∈I} |
An indexed set, the set contains all numbers pi with i being the element of some set.
{x|∃i∈I(x=pi)} |
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What is it?
Re-express: {x|∃i∈I(x=pi)} |
An indexed set, the set contains all numbers pi with i being the element of some set.
{pi|i∈I} |
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An indexed set, the set contains all numbers pi with i being the element of some set.
Give two ways |
{x|∃i∈I(x=pi)}
{pi|i∈I} |
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Re-express the family of sets F in bracket notation:
using the classes of students Cs give two ways |
Each Cs is a set itself
{{1 2},{3,4},{5,6}} C1 = {1,2} F = {Cs|s∈S} {X|∃s∈S(X=Cs)} |
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What is it?
Re-express: F = {Cs|s∈S} |
Indexed Family
Each Cs is a set itself {{1 2},{3,4},{5,6}} C1 = {1,2} {X|∃s∈S(X=Cs)} |
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What is it?
Re-express: F = {X|∃s∈S(X=Cs)} |
Indexed Family
Each Cs is a set itself {{1 2},{3,4},{5,6}} C1 = {1,2} F = {Cs|s∈S} |
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What is it?
Re-express P(A) |
The power set:
{x|x⊆A} x is a set |
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What is it?
Re-express {x|x⊆A} x is a set |
The power set:
P(A) |
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The family set is a subset of what?
Give reasoning |
If all courses were set C then each Cs is a subset of C
For each student Cs∈P(C) Every element then of the family is an element of P(c) so F ⊆ P(C) |
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Equivalence
P(A⋂B) |
P(A)⋂P(B)
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Equivalence
P(A)⋂P(B) |
P(A⋂B)
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Explain and define:
⋂F |
F = {{1,2},{2,3},{3,4}}
this is the intersection of the family {1,2}⋂{2,3}⋂{3,4} {x|∀A∈F(x∈A)} {x|∀A(A∈F → x∈A)} |
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What is this?/
{x|∀A∈F(x∈A)} |
F = {{1,2},{2,3},{3,4}}
this is the intersection of the family {1,2}⋂{2,3}⋂{3,4} {x|∀A∈F(x∈A)} {x|∀A(A∈F → x∈A)} |
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What is this?
{x|∀A(A∈F → x∈A)} |
F = {{1,2},{2,3},{3,4}}
this is the intersection of the family {1,2}⋂{2,3}⋂{3,4} {x|∀A∈F(x∈A)} {x|∀A(A∈F → x∈A)} |
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What is this?
{1,2}⋂{2,3}⋂{3,4} |
F = {{1,2},{2,3},{3,4}}
this is the intersection of the family {1,2}⋂{2,3}⋂{3,4} {x|∀A∈F(x∈A)} {x|∀A(A∈F → x∈A)} |
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Explain and define:
⋃F |
F = {{1,2},{2,3},{3,4}}
this is the union of the family {1,2}⋃{2,3}⋃{3,4} {x|∃A∈F(x∈A)} {x|∃A(A∈F ^ x∈A)} |
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What is this?/
{x|∃A∈F(x∈A)} |
F = {{1,2},{2,3},{3,4}}
this is the union of the family {1,2}⋃{2,3}⋃{3,4} {x|∃A∈F(x∈A)} {x|∃A(A∈F ^ x∈A)} |
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What is this?
{x|∃A(A∈F ^ x∈A)} |
F = {{1,2},{2,3},{3,4}}
this is the union of the family {1,2}⋃{2,3}⋃{3,4} {x|∃A∈F(x∈A)} {x|∃A(A∈F ^ x∈A)} |
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What is this?
{1,2}⋃{2,3}⋃{3,4} |
F = {{1,2},{2,3},{3,4}}
this is the union of the family {1,2}⋃{2,3}⋃{3,4} {x|∃A∈F(x∈A)} {x|∃A(A∈F ^ x∈A)} |
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Define
⋂F=∅ |
undefined
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What is it?
Redefine ⋂(i∈I) Ai |
Where F puts Ai as sets in a set, this is more like the union of all the sets, it is the same thing as ⋂F
{x|∀i∈I(x∈Ai)} {x|∀i(i∈I → x∈Ai)} It asks that all the sets have the same element |
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What is it?
{x|∀i∈I(x∈Ai)} |
⋂(i∈I) Ai
Where F puts Ai as sets in a set, this is more like the union of all the sets, it is the same thing as ⋂F {x|∀i∈I(x∈Ai)} {x|∀i(i∈I → x∈Ai)} It asks that all the sets have the same element |
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What is it?
Redefine ⋃(i∈I) Ai |
Where F puts Ai as sets in a set, this is more like the union of all the sets, it is the same thing as ⋃F
{x|∃i∈I(x∈Ai)} {x|∃i(i∈I ^ x∈Ai)} That at least one of the sets of Ai have the element |
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What is it?
Redefine {x|∃i∈I(x∈Ai)} |
Where F puts Ai as sets in a set, this is more like the union of all the
⋃(i∈I) Ai How is this different from a indexed set or indexed family? Because it's an element, not equal sets, it is the same thing as ⋃F {x|∃i∈I(x∈Ai)} {x|∃i(i∈I ^ x∈Ai)} That at least one of the sets of Ai have the element |
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What is it?
Define AxB |
Cartesian Product
{(a,b)| a∈A ^ b∈B} results in all ordered pairs |
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What is it?
Define {(a,b)| a∈A ^ b∈B} |
Cartesian Product
AxB results in all ordered pairs |
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Equivalence:
Ax(B⋂C) |
(AxB)⋂(AxC)
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Equivalence:
(AxB)⋂(AxC) |
Ax(B⋂C)
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Equivalence:
Ax(B⋃C) |
(AxB)⋃(AxC)
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Equivalence:
(AxB)⋃(AxC) |
Ax(B⋃C)
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Equivalence:
(AxB)⋂(CxD) |
(A⋂C)x(B⋂D)
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Equivalence:
(A⋂C)x(B⋂D) |
(AxB)⋂(CxD)
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Subset:
(AxB)⋃(CxD) ⊆ |
(A⋃C)x(B⋃D)
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Equivalence:
Ax∅ |
∅xA
∅ |
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AxB = BxA if and only if
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A =∅ or
B = ∅ or A=B |
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Truth set of P(x,y)
two versions |
{(a,b)∈AxB | P(x,y)}
{(a,b) | (a,b)∈AxB ^ P(x,y)} |
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{(a,b)∈AxB | P(x,y)}
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Truth set of P(x,y)
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{(a,b) | (a,b)∈AxB ^ P(x,y)}
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Truth set of P(x,y)
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Define a relation from A to B
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R ⊆ AxB
does not need to imply a truth set for R, it can be any subset |
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What is this?
R ⊆ AxB |
A relation
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Domain of R for AxB
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{a∈A | ∃b∈B((a,b)∈R}
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{a∈A | ∃b∈B((a,b)∈R}
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Domain of R
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Range of R
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{b∈B | ∃a∈A((a,b)∈R}
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{b∈B | ∃a∈A((a,b)∈R}
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Range of R
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Inverse of R R-1
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{(b,a)∈BxA | (a,b)∈R}
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{(b,a)∈BxA | (a,b)∈R}
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Inverse of R R-1
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Composition of two relations
R⊆AxB S⊆BxC |
SoR
{(a,c)∈AxC | ∃b∈B((a,b)∈R ^ (b,c)∈S} |
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{(a,c)∈AxC |
∃b∈B((a,b)∈R ^ (b,c)∈S} |
Composition of two relations
R⊆AxB S⊆BxC SoR |
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(r,s)∈L-1 if and only if
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(s,r)∈L
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(s,p)∈ToE if and only if
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∃c( (s,c)∈E ^ (c,p) ∈ T
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(x,y) ∈ R
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xRy
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xRy
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(x,y) ∈ R
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Relation equivalence:
(R-1)-1 |
R
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Relation equivalence:
Relation equivalence:Dom(R-1) |
Ran(R)
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Relation equivalence:
Ran(R) |
Dom(R-1)
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Relation equivalence:
Ran(R-1) |
Dom(R)
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Relation equivalence:
Dom(R) |
Ran(R-1)
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Relation equivalence:
To(SoR) |
(ToS)oR
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Relation equivalence:
(ToS)oR |
To(SoR)
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(SoR)-1
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R-1 o S-1
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Equivalence Relation:
R-1 o S-1 |
(SoR)-1
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the relation R is reflexive on A (or just reflexive if A is clear from context) if
(2 versions) |
∀x∈A(xRx)
or ∀x∈A((x,x)∈Rx) |
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the relation R is symmetric if
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∀x∈A∀y∈A(xRy → yRx)
or ∀x∈A∀y∈A((x,y)∈R → (y,x)∈R) |
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The relation R is transitive if
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∀x∈A∀y∈A∀z∈A
((xRy ^ yRz) → xRz) also can be done without shorthand |
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R is reflexive iff
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i(A) ⊆ R, where as before i(A) is the identity relation on A
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R is symmetric iff
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R=R-1
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R is transitive iff
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RoR ⊆ R
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∃!xP(x)
4 definitions |
∃x(P(x)^~∃y(P(y) ^ y≠x))
∃x(P(x)^∀y(P(y)→y=x)) ∃xP(x) ^ ∀y∀z[(P(y)^P(z))→y=z] Existence Uniqueness ∃x∀y(P(y)↔y=x) |