aarjaneiro/planetmath huggingface dataset:
Sharam Kohan

Sharam Kohan

By Mary Vahedi

Sharam Kohan has won two gold medals in National Mathematical Olympiads. The Minister of Science and Technology recognized Mr. Kohan’s innovative contributions to science with the prestigious Brilliant Mind Award. He received a D.Phil. in economics and social intervention from University of Oxford. Mr. Kohan continued his studies at Oxford University leading to a Bachelor of Jurisprudence degree and later at Harvard Law School where he received his Doctor of Juridical Science degree.

Title

Sharam Kohan

Canonical

name

SharamKohan

Date of creation

2015-11-14 15:28:51

Last modified on

2015-11-14 15:28:51

Owner

MaryamVahedi (27262)

Last modified by

MaryamVahedi (27262)

Numerical id

17

Author

MaryamVahedi (27262)

Entry type

Definition

Classification

msc 03A05

Generated on Thu Feb 8 19:21:32 2018 by

LaTeXML
Bertrand’s conjecture

Bertrand’s conjecture

Bertrand conjectured that for every

positive integer

n

>

1

, there exists at least one prime

p

satisfying

n

<

p

<

2

⁢

n

. This result was proven in 1850 by Chebyshev, but the phrase “Bertrand’s Conjecture” remains in the literature.

Title

Bertrand’s conjecture

Canonical name

BertrandsConjecture

Date of creation

2013-03-22 11:46:56

Last modified on

2013-03-22 11:46:56

Owner

KimJ (5)

Last modified by

KimJ (5)

Numerical id

13

Author

KimJ (5)

Entry type

Theorem

Classification

msc 03B05

Classification

msc 03B10

Classification

msc 11N05

Synonym

Bertrand’s postulate

Generated on Thu Feb 8 19:21:50 2018 by

LaTeXML
CNF

CNF

A

propositional formula

is a CNF

formula

, meaning

Conjunctive Normal Form

, if it is a

conjunction

of

disjunction

of

literals

(a literal is a

propositional variable

or its

negation

). Hence, a CNF is a formula of the form:

K

1

∧

K

2

∧

…

∧

K

n

, where each

K

i

is of the form

l

i

⁢

1

∨

l

i

⁢

2

∨

…

∨

l

i

⁢

m

for literals

l

i

⁢

j

and some

m

(which can vary for each

K

i

).

Example:

(

x

∨

y

∨

¬

⁢

z

)

∧

(

y

∨

¬

⁢

w

∨

¬

⁢

u

)

∧

(

x

∨

v

)

.

Title

CNF

Canonical name

CNF

Date of creation

2013-03-22 14:02:35

Last modified on

2013-03-22 14:02:35

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

7

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 03B05

Synonym

conjunctive normal form

Related topic

DNF

Related topic

AtomicFormula

Generated on Thu Feb 8 19:21:52 2018 by

LaTeXML
completeness theorem for propositional logic

completeness theorem for propositional logic

The

completeness theorem

of

propositional logic

is the statement that a wff is

tautology

iff it is a

theorem

. The if part of the statement is the soundness theorem, and the only if part is the completeness theorem. We will prove the two parts separately here. We begin with the easier one:

Theorem 1

.

Propositional logic is sound with respect to

truth-value semantics

.

Proof.

Basically, we need to show that every axiom is a tautology, and that the

inference rule

modus ponens

preserves truth. Since theorems are deduced from axioms and by applications of modus ponens, they are tautologies as a result.

Using

truth tables

, one easily verifies that every axiom is true (under any valuation). For example, if the axiom is of the form

A

→

(

B

→

A

)

, then we have

∎

Before proving the completeness portion of the theorem, we need the following

Theorem 2

.

Propositional logic is

complete

with respect to truth-value semantics.

Title

completeness theorem for propositional logic

Canonical name

CompletenessTheoremForPropositionalLogic

Date of creation

2013-03-22 19:32:44

Last modified on

2013-03-22 19:32:44

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

4

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 03B05

Generated on Thu Feb 8 19:22:15 2018 by

LaTeXML
contradictory statement

contradictory statement

A

contradictory statement

is a statement (or form) which is false due to its logical form rather than because of the meaning of the terms employed.

In

propositional logic

, a

contradictory statement

, a.k.a.

contradiction

, is a statement which is false regardless of the truth values of the substatements which form it.  According to G. Peano, one may generally denote a contradiction with the symbol

⋏

.

For a simple example, the statement

P

∧

¬

⁢

P

is a contradiction for any statement

P

.

The

negation

¬

⁢

Q

of every contradiction

Q

is a

tautology

, and vice versa:

¬

⋏

=

⋎

,

¬

⋎

=

⋏

To test a given statement or form to see if it is a contradiction, one may construct its

truth table

.  If it turns out that every value of the last column is “F”, then the statement is a contradiction.

Cf. the entry “contradiction (

http://planetmath.org/Contradiction

)”.

Title

contradictory statement

Canonical name

ContradictoryStatement

Date of creation

2013-03-22 16:27:07

Last modified on

2013-03-22 16:27:07

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Definition

Classification

msc 03B05

Synonym

contradiction

Related topic

Tautology

Related topic

LogicalConnective

Related topic

Contradiction

Generated on Thu Feb 8 19:22:40 2018 by

LaTeXML
contrapositive

contrapositive

Given an

implication

of the form

p

⟹

q

(“p implies q”) the

contrapositive

of this implication is

¬

⁢

q

⟹

¬

⁢

p

(“not q implies not p”).

An implication and its contrapositive are

equivalent

statements. When proving a

theorem

, it is often more convenient or more intuitive to prove the contrapositive instead.

Title

contrapositive

Canonical

name

Contrapositive

Date of creation

2013-03-22 11:59:17

Last modified on

2013-03-22 11:59:17

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

9

Author

yark (2760)

Entry type

Definition

Classification

msc 03B05

Generated on Thu Feb 8 19:22:42 2018 by

LaTeXML
converse

converse

Let a statement be of the form of an

implication

If

p

then

q

i.e. (

http://planetmath.org/Ie

) it has a certain

premise

p

and a

conclusion

q

.  The statement in which one has interchanged the conclusion and the premise,

If

q

then

p

is the

converse

of the first.  In other words, from the former one concludes that

q

is necessary for

p

, and from the latter that

p

is necessary for

q

.

Note that the converse of an implication and the

inverse

of the same implication are

contrapositives

of each other and thus are

logically equivalent

.

If there is originally a statement which is a (true)

theorem

and if its converse also is true, then the latter can be called the

converse theorem

of the original one.  Note that, if the converse of a true theorem “If

p

then

q

” is also true, then “

p

iff

q

” is a true theorem.

For example, we know the theorem on

isosceles triangles

:

If a

triangle

contains two

congruent

(

http://planetmath.org/Congruent2

) sides, then it has two congruent angles.

There is also its converse theorem:

If a triangle contains two congruent angles, then it has two congruent sides.

Both of these

propositions

are true, thus being theorems (see the entries angles of an isosceles triangle and determining from angles that a triangle is isosceles).  But there are many (true) theorems whose converses are not true, e.g. (

http://planetmath.org/Eg

):

If a

function

is

differentiable

on an

interval

I

, then it is

continuous

(

http://planetmath.org/ContinuousFunction

) on

I

.

Title

converse

Canonical name

Converse

Date of creation

2013-03-22 17:13:37

Last modified on

2013-03-22 17:13:37

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

24

Author

pahio (2872)

Entry type

Definition

Classification

msc 03B05

Classification

msc 03F07

Related topic

ExamplesOfContrapositive

Related topic

DifferntiableFunction

Related topic

Inverse6

Related topic

ConverseOfEulersHomogeneousFunctionTheorem

Defines

converse theorem

Defines

conversely

Generated on Thu Feb 8 19:22:44 2018 by

LaTeXML
DNF

DNF

A

propositional formula

is a DNF

formula

, meaning

Disjunctive Normal Form

, if it is a

disjunction

of

conjunctions

of

literals

(a literal is a

propositional variable

or its

negation

). Hence, a DNF is a formula of the form:

K

1

∨

K

2

∨

…

∨

K

n

, where each

K

i

is of the form

l

i

⁢

1

∧

l

i

⁢

2

∧

…

∧

l

i

⁢

m

for literals

l

i

⁢

j

and some

m

which can vary for each

K

i

.

Example:

(

x

∧

y

∧

¬

⁢

z

)

∨

(

y

∧

¬

⁢

w

∧

¬

⁢

u

)

∨

(

x

∧

v

)

.

Title

DNF

Canonical name

DNF

Date of creation

2013-03-22 14:14:08

Last modified on

2013-03-22 14:14:08

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

4

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 03B05

Synonym

disjunctive normal form

Related topic

CNF

Related topic

AtomicFormula

Generated on Thu Feb 8 19:22:45 2018 by

LaTeXML
differential propositional calculus : examples

differential propositional calculus : examples

Forthcoming

Title

differential propositional calculus

: examples

Canonical name

DifferentialPropositionalCalculusExamples

Date of creation

2013-03-22 18:07:33

Last modified on

2013-03-22 18:07:33

Owner

Jon Awbrey (15246)

Last modified by

Jon Awbrey (15246)

Numerical id

4

Author

Jon Awbrey (15246)

Entry type

Example

Classification

msc 03B05

Classification

msc 03B42

Classification

msc 03B44

Classification

msc 34G99

Classification

msc 39A12

Classification

msc 53A40

Related topic

DifferentialLogic

Related topic

PropositionalCalculus

Generated on Thu Feb 8 19:22:47 2018 by

LaTeXML
disjunction

disjunction

A

disjunction

is true if either of its

parameters

(called disjuncts) are true. Disjunction does

not

correspond to “or” in English (see

exclusive or

.) Disjunction uses the symbol

∨

or sometimes

+

when taken in algebraic

context

. Hence, disjunction of

a

and

b

would be written

a

∨

b

or

a

+

b

The

truth table

for disjunction is

a

b

a

∨

b

F

F

F

F

T

T

T

F

T

T

T

T

Title

disjunction

Canonical

name

Disjunction

Date of creation

2013-03-22 11:54:40

Last modified on

2013-03-22 11:54:40

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

14

Author

akrowne (2)

Entry type

Definition

Classification

msc 03B05

Synonym

logical or

Synonym

disjunctive truth function

Related topic

Conjunction

Related topic

PropositionalLogic

Generated on Thu Feb 8 19:22:49 2018 by

LaTeXML
equivalent formulation of substitutability

equivalent formulation of substitutability

Proposition 1

.

Suppose a

variable

x

occurs free

in a wff

A

. A term

t

is

free for

x

in

A

iff no variables in

t

are bound by a

quantifier

in

A

⁢

[

t

/

x

]

.

Proof.

We do

induction

on the complexity of

A

.

•

If

A

is atomic, then any

t

is free for

x

in

A

, and clearly

A

⁢

[

t

/

x

]

is just

A

, which has no

bound variables

.

•

If

A

is of the form

B

→

C

, then

t

is free for

x

in

A

iff

t

is free for

x

in both

B

and

C

iff no variables in

t

are bound in either

B

or

C

iff no variables in

t

are bound in

A

.

•

Finally, suppose

A

is of the form

∃

y

⁢

B

. Since

x

is free in

A

,

x

is not

y

, and

t

is free for

x

in

A

iff

y

is not in

t

and

t

is free for

x

in

B

iff, by induction,

y

is not in

t

and no variables of

t

are bound in

B

⁢

[

t

/

x

]

iff no variables of

t

are bound in

∃

y

⁢

B

⁢

[

t

/

x

]

, which is just

A

⁢

[

t

/

x

]

(since

x

≠

y

).

∎

If

x

does not occur free in

A

(either

x

occurs bound in

A

or not at all in

A

), then

t

is obviously free for

x

in

A

, but

A

⁢

[

t

/

x

]

is just

A

, and there is no guarantee that variables in

t

are bound in

A

or not.

In the special case where

t

is a variable

y

, we see that

y

is free for

x

in

A

iff

y

is not bound in

A

⁢

[

y

/

x

]

, provided that

x

occurs free in

A

. In other words,

y

is free for

x

in

A

iff no free occurrences of

x

in

A

are in the scope of

Q

⁢

y

, where

Q

is either

∃

or

∀

. So if

y

is not bound in

A

,

y

is free for

x

in

A

, regardless of whether

x

is free or bound in

A

. Also,

x

is always free for

x

in

A

.

Title

equivalent formulation of substitutability

Canonical

name

EquivalentFormulationOfSubstitutability

Date of creation

2013-03-22 19:35:57

Last modified on

2013-03-22 19:35:57

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B05

Classification

msc 03B10

\@unrecurse

Generated on Thu Feb 8 19:22:59 2018 by

LaTeXML
every proposition is equivalent to a proposition in DNF

every proposition is equivalent to a proposition in DNF

Theorem

.

Given any

proposition

, there exists a proposition in

disjunctive normal form

which is

equivalent

to that proposition.

Proof.

Any two propositions are equivalent if and only if they determine the same

truth function

. Therefore, if one can exhibit a

mapping

which assigns to a given truth function

f

a proposition in disjunctive normal form such that the truth function of this proposition is

f

, the

theorem

follows immediately.

Let

n

denote the number of

arguments

f

takes. Define

V

⁢

(

f

)

=

{

X

∈

{

T

,

F

}

n

|

f

⁢

(

X

)

=

T

}

For every

X

∈

{

T

,

F

}

n

, define

L

i

⁢

(

X

)

:

{

T

,

F

}

n

→

{

T

,

F

}

as follows:

L

i

(

X

)

(

Y

)

=

{

Y

i

X

i

=

T

¬

⁢

Y

i

X

i

=

F

Then, we claim that

f

⁢

(

Y

)

=

⋀

X

∈

V

⁢

(

f

)

⋁

i

=

1

n

L

i

⁢

(

X

)

⁢

(

Y

)

On the one hand, suppose that

f

⁢

(

Y

)

=

T

for a certain

Y

∈

{

T

,

F

}

n

. By

definition

of

V

⁢

(

f

)

, we have

Y

∈

V

⁢

(

f

)

. By definition of

L

i

, we have

L

i

(

Y

)

(

Y

)

=

{

Y

i

Y

i

=

T

¬

⁢

Y

i

Y

i

=

F

In either case,

L

i

⁢

(

Y

)

⁢

(

Y

)

=

T

. Since a

conjunction

equals

T

if and only if each term of the conjunction equals

T

, it follows that

⋁

i

=

1

n

L

i

⁢

(

Y

)

⁢

(

Y

)

=

T

. Finally, since a

disjunction

equals

T

if and only if there exists a term which equals

T

, it follows the right hand side equals equals

T

when the left-hand side equals

T

.

On the one hand, suppose that

f

⁢

(

Y

)

=

F

for a certain

Y

∈

{

T

,

F

}

n

. Let

X

be any element of

V

⁢

(

f

)

. Since

Y

∉

V

⁢

(

f

)

, there must exist an index

i

such that

X

i

≠

Y

i

. For this choice of

i

,

Y

i

=

¬

⁢

X

i

Then we have

L

i

(

X

)

(

Y

)

=

{

¬

⁢

X

i

X

i

=

T

¬

⁢

¬

⁢

X

i

X

i

=

F

In either case,

L

i

⁢

(

X

)

⁢

(

Y

)

=

F

. Since a conjunction equals

F

if and only if there exists a term which evaluates to

F

, it follows that

⋁

i

=

1

n

L

i

⁢

(

X

)

⁢

(

Y

)

=

F

for all

X

∈

V

⁢

(

f

)

. Since a disjunction equals

F

if and only if each term of the conjunction equals

F

, it follows that the right hand side equals equals

F

when the left-hand side equals

F

.

∎

Title

every proposition is equivalent to a proposition in DNF

Canonical

name

EveryPropositionIsEquivalentToAPropositionInDNF

Date of creation

2013-03-22 15:06:58

Last modified on

2013-03-22 15:06:58

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

10

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 03B05

Generated on Thu Feb 8 19:23:02 2018 by

LaTeXML
example of tautology

example of tautology

An example of a

tautology

and how to test it is given in the

truth table

below for the statement

(

P

∨

Q

)

∨

(

¬

⁢

P

∧

¬

⁢

Q

)

P

Q

¬

⁢

P

∧

¬

⁢

Q

P

∨

Q

(

P

∨

Q

)

∨

(

¬

⁢

P

∧

¬

⁢

Q

)

F

F

T

F

T

F

T

F

T

T

T

F

F

T

T

T

T

F

T

T

Thus for whatever truth values P and Q take on, the statement always comes out true as shown in the last coloumn of the truth table.

Title

example of tautology

Canonical name

ExampleOfTautology

Date of creation

2013-03-22 15:27:38

Last modified on

2013-03-22 15:27:38

Owner

bloftin (6104)

Last modified by

bloftin (6104)

Numerical id

4

Author

bloftin (6104)

Entry type

Example

Classification

msc 03B05

Classification

msc 03B10

Generated on Thu Feb 8 19:23:08 2018 by

LaTeXML
examples of contrapositive

examples of contrapositive

Recall that the

contrapositive

of an

implication

p

⟹

q

is the

equivalent

implication

¬

⁢

q

⟹

¬

⁢

p

, which is read: “not

q

implies not

p

”. The following are examples of the contrapositive and

converse

of a logical statement:

1.

Let

p

be the statement “it is raining” and let

q

be “the ground is getting wet”. Then the statement “if it is raining then the ground is getting wet” is equivalent to “if the ground is not getting wet then it is not raining”. Notice that these are both true statements. Notice also that the converse would be “if the ground is getting wet then it is raining” (which is not necessarily true!).

2.

Let

f

:

S

→

T

be a

function

of sets and let

S

be finite. The contrapositive statement of “if

f

is

surjective

then

T

is finite” (a true statement) would be the implication “if

T

is not finite then

f

is not surjective” (also a true statement). The converse would be “if

T

is finite then

f

is surjective” (a false statement).

Title

examples of contrapositive

Canonical name

ExamplesOfContrapositive

Date of creation

2013-03-22 16:23:05

Last modified on

2013-03-22 16:23:05

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

6

Author

alozano (2414)

Entry type

Example

Classification

msc 03B05

Related topic

ConverseTheorem

Generated on Thu Feb 8 19:23:09 2018 by

LaTeXML
exclusive or

exclusive or

The logical operator xor, short for “

exclusive or

”, is a

binary operator

that is true iff exactly one of its

parameters

is true, and thus corresponds to the normal use of “or” in English. Xor uses the symbol

⊕

due to its similarity to

disjunction

, which sometimes uses

+

. The

truth table

for xor is

a

b

a

⊕

b

F

F

F

F

T

T

T

F

T

T

T

F

Title

exclusive or

Canonical

name

ExclusiveOr

Date of creation

2013-03-22 18:04:59

Last modified on

2013-03-22 18:04:59

Owner

me_and (17092)

Last modified by

me_and (17092)

Numerical id

6

Author

me_and (17092)

Entry type

Definition

Classification

msc 03B05

Synonym

xor

Synonym

logical xor

Synonym

logical exclusive or

Related topic

Disjunction

Related topic

PropositionalLogic

Generated on Thu Feb 8 19:23:11 2018 by

LaTeXML
functional completeness

functional completeness

Recall that in classical

propositional logic

,

well-formed formulas

(wffs) can be built up (recursively) from

propositional variables

via

logical connectives

. There are several choices for the logical connectives used:

•

F

1

=

{

¬

,

∨

}

,

•

F

2

=

{

¬

,

∧

}

,

•

F

3

=

{

¬

,

→

}

,

•

F

4

=

{

¬

,

∨

,

∧

}

,

•

F

5

=

{

¬

,

∨

,

∧

,

→

,

↔

}

.

For a given set

V

of (propositional)

variables

, and a set

F

of logical connectives, denote

V

¯

⁢

(

F

)

the set of all wffs built from

V

with respect to

F

. From the choices above, we see that

V

¯

⁢

(

F

i

)

⊂

V

¯

⁢

(

F

5

)

for all

i

<

5

, and

V

¯

⁢

(

F

j

)

⊂

V

¯

⁢

(

F

4

)

for all

j

<

3

.

However, we know that, intuitively, some of the

connectives

are “redundant” in that they can be “defined” using existing connectives. For example, the connective

↔

can be defined in terms of

→

and

∨

:

p

↔

q

:=

(

p

→

q

)

∨

(

q

→

p

)

,

and

→

can in turn be defined in terms of

∨

and

¬

:

p

→

q

:=

(

¬

⁢

p

)

∨

q

,

etc… This means that, although

V

¯

⁢

(

F

5

)

is a much larger set than, say,

V

¯

⁢

(

F

1

)

, every extra wff in

V

¯

⁢

(

F

5

)

is in some way

equivalent

to an wff in

V

¯

⁢

(

F

1

)

. This equivalence is the familiar semantic equivalence. If fact, we can show that

¬

and

∨

are all we need: “any” logical connective can be “defined” in terms of them, not just the ones mentioned above. This is the notion of

truth functional completeness

, or

functional completeness

for short. To make this precise, we have the following:

Definition

A set

F

of logical connectives is said to be

truth functionally complete

, or

functionally complete

if, given logical connective

ϕ

, every wff in

V

¯

⁢

(

F

∪

{

ϕ

}

)

is

semantically equivalent

to a wff in

V

¯

⁢

(

F

)

, considered as a subset of

V

¯

⁢

(

F

∪

{

ϕ

}

)

.

It is clear that if

F

is functionally complete, so is any of its

superset

. Also, given a set

F

of logical connectives, if there is a functionally complete set

G

of logical connectives such that every wff in

V

¯

⁢

(

G

)

is semantically equivalent to a wff in

V

¯

⁢

(

F

)

, then

F

is functionally complete.

For example, it can be shown that

F

1

above is functionally complete, and as an easy

corollary

, so is each of the rest of

F

i

above.

References

1

D. van Dalen:

Logic and

Structure

, Springer, 4th Ed., Berlin (2008).

2

H. Enderton:

A Mathematical

Introduction

to Logic

, Academic Press, San Diego (1972).

Title

functional completeness

Canonical

name

FunctionalCompleteness

Date of creation

2013-03-22 18:51:32

Last modified on

2013-03-22 18:51:32

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B05

Synonym

truth functional completeness

Synonym

truth functionally complete

Defines

functionally complete

Generated on Thu Feb 8 19:23:22 2018 by

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implication

implication

An

implication

is a logical construction that essentially tells us if one condition is true, then another condition must be also true. Formally it is written

a

→

b

or

a

⇒

b

which would be read “

a

implies

b

”, or “

a

therefore

b

”, or “if

a

, then

b

” (to name a few).

Implication is often confused for “if and only if”, or the

biconditional

truth function

(

⇔

). They are not, however, the same. The implication

a

→

b

is true even if only

b

is true. So the statement “pigs have wings, therefore it is raining today”, is true if it is indeed raining, despite the fact that the first item is false.

In fact, any implication

a

→

b

is called

vacuously true

when

a

is false. By contrast,

a

⇔

b

would be false if either

a

or

b

was by itself false (

a

⇔

b

≡

(

a

∧

b

)

∨

(

¬

⁢

a

∧

¬

⁢

b

)

, or in terms of implication as

(

a

→

b

)

∧

(

b

→

a

)

).

It may be useful to remember that

a

→

b

only tells you that it

cannot

be the case that

b

is false while

a

is true;

b

must “follow” from

a

(and “false” does follow from “false”). Alternatively,

a

→

b

is in fact

equivalent

to

b

∨

¬

⁢

a

The

truth table

for implication is therefore

a

b

a

→

b

F

F

T

F

T

T

T

F

F

T

T

T

Title

implication

Canonical

name

Implication

Date of creation

2013-03-22 11:53:00

Last modified on

2013-03-22 11:53:00

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

10

Author

akrowne (2)

Entry type

Definition

Classification

msc 03B05

Classification

msc 81T70

Classification

msc 81T60

Synonym

conditional truth function

Related topic

PropositionalLogic

Defines

vacuously true

Defines

implies

Generated on Thu Feb 8 19:23:24 2018 by

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