Learning Logical equivalences is a type of relationship between any two reports. If p and q are the two reports if they are logically equal mean p and q are same. The logical relation of any two reports is related by the report "if and if only". Now we are going to learn how to solve the logical equivalences. For this we have to learn the logical equivalences laws and how we are indicating the logical equivalence. Because to solve the logical equivalence we have to learn the laws and we have to apply it. In this article we shall discuss about learning logical equivalence.

Laws of Logical Equivalences:

We have to learn the following laws to solve the Learning logical equivalence

De Morgan"s laws
Distributive law and prepositions
Complement laws
Identity laws
Commutative laws
Associative laws

The logical Equivalence is indicated by " OR =

Learning De Morgan's Law:

Law for logical in equivalence

(P V Q) "" (P) ^ (Q)

(P ^ Q) "" (P) V (

Learning Distributive law:

Law for logical in equivalence

P^ (Q V R) = (P ^ Q) V (p ^R)

PV (Q ^ R) = (P V Q) ^ (P VR)

Learning Complement Laws:

Law for logical in equivalence

P V P " T

P ^ P " F

P" T

T " F

F " T

Learning Identity laws:

Law for logical in equivalence

P ^ T " P

P ^ F " F

P V T " T

P V T " P

Learning Commutative laws:

Law for logical in equivalence

P ^ Q " Q ^ P

P V Q " Q

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Learning Associative Laws:

Law for logical in equivalence

P ^ (Q ^ R) " (P ^ Q) ^ R

P V (Q V R) " (P V Q) V R

Applying Logical Equivalences Laws for problems:

Learning logical equivalence Problem 1:

(P ^ Q) = not (P ^ Q) V (P V Q)

Law of implication

= (not P V not Q) V (P V Q)

De Morgan's law

= (not P V P) V (not Q V Q)

Associative law

=T V T

=T

The word logic indicates an analysis. Analysis may be the approved answer or mathematical proof. The logical equivalence is one of the processes of logical statements. We use ' symbol to signify the word logical equivalence in the discrete mathematics. Let us see preparation for logical equivalence in this article.

Preparation for Logical Equivalence:

Logic:

Significance of the term logic is analysis. Analysis may be approved result or mathematical proof.

Logical statement:

Logical statement is a sentence which is any one true or false but not both.

Logical operator:

AND
OR
NOT
Logical equivalence
Conditional statement
Bi "" conditional statement
Tautology
Contradiction

Logical equivalence:

If the final column of two components are similar then that two components are said to be logical equivalence.

Some other preparation about logical equivalence:

The symbol "-= " is used for represent the term logical equivalence.
Further name of logical equivalence is simple equivalence.

Problems for Preparation for Logical Equivalence:

Problem 1:

Prove ~ (Avv B) -= (~A) ^^ (~B)

Solution:

LHS:
ABAvv B~ (Avv B)
TTTF
TFTF
FTTF
FFFT

RHS:
AB~A~B(~A) ^^ (~B)
TTFFF
TFFTF
FTTFF
FFTTT

The last column value of LHS and RHS are equal so given is simple or logical equivalence.

Problem 2:

Prove ~ (A^^ B) -= (~A)vv (~B)

Solution:

LHS:
ABA^^ B~ (A^^ B)
TTTF
TFFT
FTFT
FFFT

RHS:
AB~A~B(~A)vv (~B)
TTFFF
TFFTT
FTTFT
FFTTT

The last column value of LHS and RHS are equal so given is simple or logical equivalence.

Problem 3:

Prove (Avv B)|- (A^^ B) -= Aharr B

Solution:

LHS:
ABAvv BA^^ B(Avv B)|- (A^^ B)
TTTTT
TFTFF
FTTFF
FFFFT

RHS:
ABAharr B
TTT
TFF
FTF
FFT

The last column value of LHS and RHS are equal so given is simple or logical equivalence.