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A logical expression results from applying logical or boolean operators to expressions. Expressions may be relational or arithmetic. Parentheses can be used to group different operands with their operator. Parentheses are the round brackets left round bracket ( and the right round bracket ). Sometimes the parentheses use different shape brackets like { }, [ ]. It usually employs several different techniques. The result of an operation comes in two possible states, either true or false. Commonly used operators are AND operator, OR operator, NOT operator. Logical operators are evaluated after all operations have been evaluated.
Logical expressions:
A logical or boolean expression results from applying logical or boolean operators to expressions. Expressions may be relational or arithmetic. Parentheses can be used to group different operands with their operator. Parentheses are the round brackets left round bracket ( and the right round bracket ). Sometimes the parentheses use different shape brackets like { }, [ ]. It usually employs several different techniques. The result of an operation is either true or false. Commonly used operators are AND operator, OR operator, NOT operator. Logical operators are evaluated after all operations have been evaluated. If two or more logical operators appear in an expression, the leftmost operator will perform first.
The list of operators is as follows:-
The statements with words such as AND, OR, NOT are used for a combination of words that combines one or more mathematical statements. Special symbols are used to represent the compound statements.
AND Operator:-The conjunction of statements P and Q is denoted as P ∧ Q. The statement P ∧ Q is true when both Pand Q are true.
OR Operator:- The disjunction of statements P and Q are denoted as P V Q. The statement P V Q is true when atleast one of P or Q is true.
NOT Operator:- The negation of statements P and Q is denoted as P → Q. The statement P → Q is false only when P is true, and Q is false.
Before discussing the steps, we must be clear with Distributive Law and Tautology Law.
Distributive Law:- As per the law operations of multiplication and addition, stated symbolically, p(q+r) = pq+ pr; i.e. the monomial factor pis distributed or separately applied to each term of the binomial factor q + r, resulting in the product pq+ pr.
Tautology Law:- A tautology is a preposition, i.e. true. A proposition that is always false is called a contradiction.
The important tautologies in the theorem are as follows:-
P⇔¬¬P
P V Q⇔Q V P
P∧Q⇔Q ∧P
Truth table is a table that is a breakdown of logic function by listing all possible values. A truth table may contain several rows and columns. The top row represents the logical variables and combinations. The truth table logic gates give us all the information about the combination of inputs and outputs for the logic operation. In a truth table, a statement is called a contradiction if the final column of that truth table contains only 0’s. If the final column of that truth table contains both 0’s and 1’s this is called contingency or contingent. A truth table shows the truth values of a statement formula for every combination.
Steps to Solve Logical Expression:
Use logical laws to simplify the expression.
Use Distributive Law.
Example:- (P∧Q)V(P∧¬Q)
P∧(QVR) =(P∧Q)V(P∧R)
(P∧Q)V(P∧¬Q) = P∧(QV¬Q)
Tautology Law
P tautology = P
P∧(QV¬Q) = P.
To solve all this, a basic need is a truth table.
Conclusion:
Logical reasoning is useful for solving maths problems. One can find the conclusion based on given facts and mathematical principles. A logical expression or boolean expression is a result of applying logical or boolean operators to expressions. Expressions may be relational or arithmetic. Parentheses can be used to group different operands with their operator..Parentheses are basically the round brackets
