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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