Modus Ponen & Chain Rule

 Premise:

A premise is a proposition on which an argument is based or from which a conclusion is drawn

Hypothetical Proposition:

 It is a proposition in the form:

       

            "If P, then Q"    ( Symbolically : P→Q )

 here,

            P is known as "Antecedent"

            Q is known as "Consequent"

Inference Rules:

In Propositional Logic, there are 2 Inference Rules drawn from a hypothetical proposition:

        ➤ Modus Ponen

        ➤ Modus Tollen

Modus Ponen:    It is a method of affirming the consequent. This inference rule (modus ponen) form resembles a syllogism with 2 premises and a conclusion:

                            If P, then Q     (Symbolically : P→Q )    ( Premise 1)
                            

                            P,                                                               ( Premise 2)
            therefore,

                            Q                                                               ( Conclusion)

we can understand it with this example:

                            "If you have an ATM card, then you can withdraw money from ATM machine"
     

                             You have an ATM card
                 therefore,

                             You can withdraw money from ATM machine.

Modus Tollen:    It is a method of denying the antecedent. It is an argument of the form:

                            If P, then Q     (Symbolically : P→Q )

                           ~Q,
           therefore,

                            ~P

we can understand it with following example:

                    "If you have an ATM card, then you can withdraw money from ATM machine"
     

                      You can not withdraw money from ATM machine
             therefore,

                      you do not have an ATM card.

Chain Rule:    It is hypothetical syllogism with a conditional for one or both of its premises.

we can express it algebrically as:

                        P→Q

                        Q→R

        therefore,

                        P→R

Let us understand it with this example:

                        "If I go to school daily, then I will be well educated"

                        "If I will be well educated, then I will get a good job"

from this we can conclude,

                        "If I go to school daily, then I will get a good job"

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Propositional Logic

Propositions are simple atomic sentences which are connected using logical connectives and propositional logic represents logic through propositions

A proposition can have true or False as its value.

This part will cover the propositions, their types and other related terms.

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Conditional and Bi-Conditional Laws

 Conditonal Law:

This law is also known as Law of Inference or Implication. For any 2 binary values P & Q, it represents the "If P then Q" relationship.

which means that for any set of truth values, if P is true than Q should be true to generate a true result otherwise it will give you false. Rest other cases will be true.

This law is represented as :

P→ Q           If P, then Q, here P is known as "Antecedent" & Q as "Consequence"
OR
P⇒ Q
OR
P ⊃ Q

It can be algebrically represented as P'+Q,.

Same can be seen in the following truth table that when P=1 & Q=0 then, P→Q results in 0(False), rest all cases are 1(True).

P	Q	P→Q
0	0	1
0	1	1
1	0	0
1	1	1

Bi-Conditional Law:

This law is also known as Law of Equivalence. For any 2 binary values P & Q, it represents the "If and only if " relationship.

which means that for any set of truth values, if and only if both are same than it generates a True result. Rest other cases will be False.

This law is represented as :

P↔Q           If and only if both are same

OR
P⇔Q

OR
P ≡ Q

It can be algebrically represented as P'Q' + PQ

Same can be seen in the following truth table that when P=0 & Q=0 OR when P=1 & Q=1

then P⇔Q results in 1(True), rest all cases are 0(False).

P	Q	P⇔Q
0	0	1
0	1	0
1	0	0
1	1	1

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Truth Table

Truth tables are basically tables that contains all the possible set of truth values of all inputs and represent a boolean expression in table form.

These tables are useful in evaluating a boolean expression for a particular set of truth values.

these tables make it simple and easy to prove various laws and relation between boolean expressions.

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Maxterms And POS Expressions

Maxterm (M):

Maxterms are formed when binary inputs are connected using OR(+). They are denoted by "M".

example: A'+B, A+B'+C, A+B'+C'

POS (Product of Sums) Expression:

POS expressions are formed by connecting maxterms using AND(.).

example: (A'+B).(A+B'+C).(A+B'+C')

Canonical POS Expressions:

Canonical POS expressions are those POS expressions which contains all inputs in its every maxterm.

example: (A'+B+C).(A+B'+C).(A+B'+C')

Cardinal (Function) Form of POS Expression:

This is the function notation of a Canonical POS expression using 𝝿(Pie).

example : F(A, B, C) = 𝝿 (0, 1, 2, 6).

This is done by writing the canonical POS expression in its hexadecimal valued maxterm.

example:  if the canonical POS expression is (A'+B+C).(A+B'+C).(A+B'+C') then its cardinal notation can be formed by:

            A'+B+C = 1+0+0 = M4

            A+B'+C = 0+1+0 = M2

            A+B'+C' = 0+1+1 = M3

    so,  F(A, B, C) = 𝝿 (2, 3, 4).

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Minterm And SOP Expression

 Minterm (m):

Minterms are formed when binary inputs are connected using AND(.). They are denoted by "m".

example: A'.B, A.B'.C, A.B'.C'

SOP (Sum of Products) Expression:

SOP expressions are formed by connecting minterms using OR(+).

example: A'.B + A.B'.C + A.B'.C'.

Canonical SOP Expressions:

Canonical SOP expressions are those SOP expressions which contains all inputs in its every minterm.

example: A'.B.C + A.B'.C + A.B'.C'.

Cardinal (Function) Form of SOP Expression:

This is the function notation of a Canonical SOP expression using 𝚺 (Sigma).

example : F(A, B, C) = 𝚺 (0, 1, 2, 6).

This is done by writing the canonical SOP expression in its hexadecimal valued minterm.

example:  if the canonical SOP expression is A'.B.C + A.B'.C + A.B'.C' then its cardinal notation can be formed by:

            A'.B.C = 0.1.1 = m3

            A.B'.C = 1.0.1 = m5

            A.B'.C' = 1.0.0 = m4

    so,  F(A, B, C) = 𝚺 (3, 4, 5).

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The Easiest Way to Study

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I am new to this Blogging world and will learn its features and rules by the time.

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