Bi Implication Inwards Mathematical Logic

Definition of Bi Implication

Two statements p together with q tin last made into a novel contestation or chemical compound judgement into the cast "p if together with solely if q". This novel contestation prepared inwards such a means is called the contestation of the bi iden- tation of the statements p together with q.

Formula Bi Implications inwards Mathematical Logic

p ↔ q

Information :
p ↔ q reads "p if together with solely if q"

Example Bi Implications inwards Mathematical Logic

p : v > 1
q : 3² = 9
p ↔ q: v > ane if together with solely if 3² = 9

Truth value tabular array of Bi Implication

The truth value of the biimplication contestation is determined every bit follows:

The p ↔ q bi implication is truthful if p together with q accept the same truth value. In other possibilities bi implications are false.

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Implications Inwards Mathematical Logic

Two statements p too q tin last made into i novel tilt of chemical compound sentences into the shape "if p hence q". New statements arranged inwards this agency are either implied statements or conditional / conditional statements of p too q statements. The "if p" role is called the argue or campaign (antecedent / hypothesis) too the "then q" is called conclution or resultant (conclusion or consequence).

Formula of Implications

p → q 

Information:
p → q reads "if p hence q"

Examples of Implications inwards Mathematical Logic

p    : It's a cloudy day
q    : Now it volition rain
p → q: If it's cloudy at in i trial it volition rain

Truth Table of Implications

The value of the truth of the implication tilt is determined past times the truth value of each percentage non past times the human relationship of the 2 sole statements. The value of truth implication is every bit follows:

The implication p → q is imitation if p is truthful too q is wrong, inwards other probability p → q is true.

Information :
T : True
F : False/wrong

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Disjunction Inward Mathematical Logic

Two statements p in addition to q tin endure combined using a hyphen "or" to shape a novel statement. This chemical compound arguing is called a disjunction. The disjunction of the p in addition to q statements is written "p ν q" in addition to read "p disjungs q" or "p or q".

In everyday life the discussion "or" agency ane or both, it tin endure either merely non both.

Formula of Disjunction

p ν q

Information :
p ν q read "p or q"

Example of Disjunction inwards Math Logic

p    : Two lines are parallel
q    : Two lines intersect
p ν q: Two lines are parallel or intersect each other

Truths value of Disjunction Table

The value of the truth of the multiple disjunction statements of 2 statements p in addition to q is determined every bit follows:

If p is truthful in addition to q is valuable or p in addition to q are both truthful in addition to thence p ν q is true. If non in addition to thence p ν q is wrong. In other words the disjunction of 2 statements is imitation alone if the 2 components are wrong.

Information :
T : True
F : False/Wrong

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Conjunction Inwards Mathematical Logic

Definition of Conjunctions inwards Mathematical Logic

Two statements p as well as q tin travel combined using the "and" hyphens to cast a chemical compound tilt called the conjunction of the p as well as q statements. The conjunction of the p as well as q statements is expressed by:

Formula or symbol of Conjunctions inwards Mathematical Logic 

p ∧ q

Information :
p ∧ q read "p as well as q"

Example of Conjunctions inwards Mathematical Logic

p : Djakarta is the Capital of Indonesia
q : Djakarta is located on the isle of Java
p ∧ q : jakarta is the uppercase of Republic of Indonesia as well as is located on the isle of Java.

Truth value of Conjunction

The truth value of the conjunctive chemical compound tilt of 2 statements p as well as q is determined equally follows :

If p is truthful as well as q is truthful hence p ∧ q is true. If 1 of the statements is faux hence p ∧ q is false.

Information :
T = True
F = False or wrong

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Statements As Well As Non Statements Inwards Math Logic

In everyday life, if you lot desire to limited something, together with thus ever purpose the judgement (a serial of words). According to the logic of the judgement system is every bit follows:

1. Meaning Sentence

Sentence agency meaningful sentence

Example of Meaning Sentence

Fatima is a 10th grader
Jakarta is located on the isle of Java
6 x 8 = 50

2. Not Meaning Sentence

Not Meaning Sentence is a judgement that has no meaning

Example of Not Meaning Sentence

The depository fiscal establishment loves eight
Three dining cabinets

3. Statement Sentence

The contestation judgement is a judgement that has truthful or imitation values only, but non both. Thus nosotros tin say that the contestation is a judgement that has truthful value entirely or incorrect asja but non truthful together with incorrect all at once, or inward other words a contestation is a judgement that tin live on determined the value of truth. To simplify subsequent use, the contestation volition live on denoted past times a small-scale letter, such every bit p, q, r together with thus on. The truthful contestation has a truthful value of T (true) or 1 together with the imitation contestation has a T (wrong) or 0 truth value.

Example of Statement Sentence

p: The counting set out is the master summation the zero
q: Republic of Indonesia Raya vocal created past times Kusbini
r: If 2x = vi together with thus x = 3

In the event of the contestation above, p together with r are truthful piece q is false.

4. Statement of facts Sentence

Statement of facts Sentence is a contestation whose truthful value should live on investigated first.

Example of Statement of facts Sentence

Hanif is ane of the students of SMK Taruna
Fulan is a corrupt person
There has been a burn at Bumi Maya Housing

5. Not Statement Sentence

Not Statement Sentence is a judgement that tin non live on determined its truthful value.

Example of Not Statement Sentence

May God forgive your sins
What is the set out of students of SMK inward Jakarta?
Rest if you lot are tired

6. Open sentence

Open judgement is a judgement containing variable (variable) together with if the variable is replaced past times a constant inward its semestation, it volition attain a statement.

Exmaple of Open sentence

x + 2 = 5
x² - 5x - twoscore > 0
This is a metal

Influenza A virus subtype H5N1 variable inward the opened upward sentence, if it is replaced together with thus the judgement tin live on determined its truthful value. Review x + 2 = 5, if x nosotros supervene upon amongst 3 together with thus the judgement becomes 3 + 2 = 5 is a judgement that is truthful together with x = 3 is called completion of the opened upward sentence. But if the cost of x nosotros supervene upon amongst 1 together with thus the judgement becomes 1 + 2 = 5, this is a contestation that is incorrect value.

From the higher upward review nosotros tin say that the opened upward judgement tin plough into a contestation that is truthful or imitation if the variable or variable of the opened upward judgement is replaced amongst a certainly value.

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Compound Statements Inwards Math Logic

Definition of Compound Statements inward Mathematical Logic

Influenza A virus subtype H5N1 chemical compound tilt or multiple judgement is a novel tilt composed of 2 or to a greater extent than statements using the discussion logic, ie and, or only in addition to then on. The tilt of the chemical compound of the chemical compound tilt is called the components or sub-statements.

Example of Compound Statements inward Mathematical Logic

Bandung the uppercase of West Java province in addition to located on the isle of Java.
Component forming chemical compound judgement is Bandung the uppercase of West Java in addition to Bandung is located inward West Java.

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Negation Inwards Mathematical Logic

Definition of Negation inwards Mathematical Logic

Negation is normally used to deny or opposite a statement. To deny or brand a negation of a disceptation is normally yesteryear putting the give-and-take "not true" inwards forepart of a judgement or yesteryear inserting not. "The novel disceptation obtained inwards such a means is called the negation of an initial statement.

Formula or Symbol of Negation inwards Mathematical Logic

If p is a statement, the negation of the disceptation tin flame last written equally follows:

p

Information :
p reads "not truthful p" or "not p"

Example Question of Negation inwards Mathematical Logic

Please possess upwards one's heed the negation of the statements:
p: DKI Jakarta upper-case missive of the alphabet of Indonesia
p: Not truthful DKI Jakarta upper-case missive of the alphabet of Indonesia
p: DKI Jakarta is non the upper-case missive of the alphabet of Indonesia

Truth Value of Negation

If p is a truthful value, in addition to therefore p is incorrect in addition to vice versa if it is imitation in addition to therefore p is true, for the truth tabular array it tin flame last seen inwards the truth tabular array below:

Information :
T = True
F = False / Wrong.

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Quantification Judgement Inwards Math Logic

An opened upward judgement tin dismiss live on converted into a controversy if the variable of the judgement is substituted past times a for sure constant.

For example:
Open sentence: x + four = iii for x ∈ R
If x = -1, in addition to then the higher upward judgement becomes a truthful value statement.
If x = 2, in addition to then the opened upward judgement becomes a controversy of imitation value.

Another agency to plow an opened upward judgement into a controversy is to purpose a Quantification

There are ii types of quantification inward mathematical logic, such as:

1. Universal Quantification
2. Existential Quantification

1. Universal Quantification

Universal Quantification is written amongst the symbol "∀" in addition to read "for all" or "for each". If p(x) is an opened upward judgement in addition to given a universal quantification it volition live on a revelation in addition to written (∀x) p(x) read:

• For each cost x apply the nature p.
• For all prices x has properties p.

The shape (∀x) p(x) is a declarative controversy that has a truth value to live on truthful or false, ie if it tin dismiss non live on flora x that is non p(x) in addition to then (∀x) p(x) is true. If x tin dismiss live on flora that is non p(x), in addition to then (∀x) p(x) is false.

Example of Universal Quantification

∀x existent number, x² = 1
This controversy is imitation fifty-fifty if it applies to x = - 1 or x = 1 ie 1² = 1 in addition to (1)² = 1 precisely does non apply to all x (eg x = 3, in addition to then 3² ≠ 1)

2. Existential Quantification

Existential Quantification is written amongst the symbol "∃" in addition to read "any / some" or "at to the lowest degree one". If p (x) is an opened upward judgement in addition to given an existential interrogation it volition live on a controversy in addition to written (∃x) p(x) read:

• there is x such that the applied properties p.
• some x convey properties p.
• at to the lowest degree i x past times nature p.

The shape (∃x) p(x) is a declarative controversy that has a truth value to live on truthful or imitation ie if it tin dismiss live on flora at to the lowest degree i x which is p(x) then (∃x) p(x) is true. If no x tin dismiss live on flora inward p(x) in addition to then (∃x) p(x) is false.

Example of Existential Quantification

∃x existent number, x < 1
The controversy is imitation because it tin dismiss non live on determined x the master copy release < 1.

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Conversions, Inversions, As Well As Contraposition Inwards Math Logic

From an implication contention p → q tin forcefulness out move made a novel statement, such as:

q → p, is called the conversions of implications
P → q, is called the Inversions of the implications
q → p, is called the contraposition of the implications

Conversion Example

For example:
p: Triangle ABC equilateral
q: The iii corners are the same
p → q: "If the triangle ABC is equal sides in addition to hence The iii corners are the same"
q → p: "If The iii corners are the same in addition to hence the triangle ABC is equal"

Invers Example

For example:
p: Triangle ABC equilateral
q: The iii corners are the same
p → q: "If the triangle ABC is non equal sides in addition to hence The iii corners are the same"
p → q: "If the triangle ABC is non equal side in addition to hence The iii corners are non the same."

Contraposition Example

For example:
p: Triangle ABC equilateral
q: The iii corners are the same
p → q: "If the triangle ABC is equal sides in addition to hence The iii corners are the same"
q → p: "If The iii corners are non the same in addition to hence the triangle ABC is non equal side"

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The Value Of Chemical Compound Statement

The truth value of a complex contestation p, q, r, ... in addition to inwards the shape of a symbol that uses contestation operations (negation, conjunction, disjunction, implication, in addition to biimplication) tin hold upward determined using a truth table.

Example of the truth tabular array of the next chemical compound contestation pairs inwards a unmarried table

a. (p ∧ q) ; p ν q

Information :
T = True
F = False

The truth value inwards column 6 is equivalent or equal to column vii is FTTT, in addition to hence it tin hold upward concluded that the negation of the p ∧ q contestation is p ν q in addition to tin hold upward written every bit follows :

(p ∧ q) ≡ p ν q

b. (p ν q) ; p ∧ q

Information :
T = True
F = False

The truth value at column 6 is equivalent or equivalent to column vii is FFFT, in addition to hence it tin hold upward concluded that the negation of the disjunction contestation p ν q is p ∧ q in addition to tin hold upward written every bit follows:

(p ν q) ≡ p ∧ q  

c. (p → q) ; p ∧ q

Information :
T = True
F = False

The truth value inwards the fifth column is equivalent or equal to the 6th column is the FTFF, in addition to hence it tin hold upward concluded that the negation of the implication contestation p → q is p ∧ q in addition to tin hold upward written every bit follows:

(p → q) ≡ p ∧ q

d. (p ↔ q); p ↔ q

Information :
T = True
F = False

The truth value inwards column 6 is equivalent or equivalent to the seventh column of FTTF, in addition to hence it tin hold upward concluded that the negation of the bi ation contestation p ↔ q is p ↔ q in addition to tin hold upward written every bit follows:

(p ↔ q); p ↔ q atau (p ↔ q); p ↔ q

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Conclusion Mathematical Logic

To bear witness the novel declaration or proof, the truth must survive shown every bit a resultant of to a greater extent than or less other grouping of statements, each of which may survive accepted every bit truthful or previously verified. The revelations received past times the truth without needing proof are called axioms. For example, "Two unlike lines tin forcefulness out non intersect at to a greater extent than than ane point".

In proving a proffer or deriving an resultant of the known truths the argumentation blueprint is used, namely past times drawing conclusions from known statements called premises based on logical principles, namely ponen mode, tollens means as well as syllogism.

Conclusions are said to survive legitimate, if the conjunctions of the premises stimulate got concluded implications. Conversely, if the conjunctions of the premises stimulate got no implication as well as so the declaration is said to survive imitation or illegitimate. Thus, a decision is said to survive valid if the premises are truthful as well as so the conclusions are likewise true.

1. Ponen mode

The ponen means is an declaration inward the shape of the following:

"If p → q is truthful as well as p is truthful as well as so q is true"

In the shape of diagrams tin forcefulness out survive presented every bit follows:

Example of Ponen Mode

Premise 1: If a kid diligently learns, as well as so he passed the test
Premise 2: Ahmad is a diligent child
Conclusion: ∴Ahmad passed the exam

To bear witness valid or non parse-drawing inference tin forcefulness out survive used truth table. Ponen means declaration "If p → q is truthful as well as q is truthful as well as so q is true" tin forcefulness out survive written inward the shape of implication, that is:

[(p → q) ∧ q] → q

This decision is said to survive valid if it is a tautology. The truth tabular array of the shape is every bit follows:

Information :
T : True
F : False

From the tabular array to a higher house it appears that [(p → q) ∧ q] → q is a tautology. So the declaration or decision of the ponen means shape is valid.

2. Tollens mode

The tollens means is an declaration inward the shape of the following:

"Jika p → q benar dan q benar maka p benar"

In the shape of diagrams tin forcefulness out survive presented every bit follows:

Example of Tollens Mode

Premise 1: If it is Sunday, as well as so Budi is on an excursion
Premise 2: Budi is non on an excursion
Conclusion: ∴ it is non Sunday

To bear witness valid or non decision past times tollens means tin forcefulness out survive used truth table. Ponen means declaration "If p → q is truthful as well as q truthful as well as so p is true" tin forcefulness out survive written inward the shape of implication, that is:

[(p → q) ∨ q] → q

This decision is said to survive valid if it is a tautology. The truth tabular array of the shape is every bit follows:

Information :
T : True
F : False

From the tabular array to a higher house it appears that [(p → q) ∨ q] → q is a tautology. So the declaration or decision of shape tollens means is valid.

3. Silogism

Silogism is an declaration shaped every bit follows:

"If p → q is truthful as well as q → r is truthful as well as so p → r is true"

In the shape of diagrams tin forcefulness out survive presented every bit follows:

Example of Silogism

Premise 1: If you lot written report hard, as well as so you lot acquire to class
Premise 2: If he goes to class, he volition purchase a bicycle
Conclusion: ∴If you lot written report hard, you lot volition purchase a bicycle

To bear witness valid or non silogism decision tin forcefulness out survive used truth table. The silogism declaration "If p → q is truthful as well as q → r is truthful as well as so p → r true" tin forcefulness out survive written inward the shape of implication, that is:

[(p → q) ∧ (q → r)] → (p → r)

This decision is said to survive valid if it is a tautology. The truth tabular array of the shape is every bit follows:

Information :
T : True
F : False

From the tabular array to a higher house it appears that [(p → q) ∧ (q → r)] → (p → r) is a tautology. So the declaration or decision of the silogism shape is valid.

The decision does non depend on the fairness or non the pregnant of the decision every bit a declaration only on the truth value of the conclusion.

• Arguments whose conclusions are meaningful only are non obtained past times using logical principles, as well as so the conclusions are invalid.
• Some of the arguments to which the conclusions are odd only are obtained past times using the principles of logic thus the conclusions are valid.

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Negation Of Quantified Statement

Negation of Universal Quantification Statement

Negation tilt "For all x applicable p(x)" is "It is non truthful that for all x use p(x)" or inwards other words "there is at to the lowest degree 1 x such that p(x) is non applicable". By using the symbol, nosotros write every bit follows:

(∀x) p(x) ≡ (∃x) p(x)

Example:
p: All cats bring a tail
p: Not actually all cats bring a tail.
p: There is a truthful cat that has no tail.
p: Some cats produce non bring tails.

Negation of Existential Quantification Statement

Negation tilt "There is x applicable p(x)" is "not truthful that at that topographic point x applies p(x)" or inwards other words "For all x such that p(x) non applicable". By using the symbol nosotros write every bit follows:

(∃x) p(x) ≡ (∀x) p(x)

Example:
p: There is a kid who likes to play ball.
p: Not truthful There is a kid who likes to play ball.
p: All children produce non similar to play ball.

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