Scratch Interface Intro

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Boolean connectives

Propositional Logic and Hardware

Connectives used in propositional logic

Negation (NOT) : Represented by ~ or ‘ or a -(bar) sign. It is an operator that affacts a single statements only and does not join two or more statements. it  it is also called as unary connective. It’s output is always inverse it input.

Conjunction (AND) : Represented by the symbol . or ^. It works on more than one proposition and the output is true when all the propositions are true. Conjunction means both the proposition are true.

Disjunction (AND) : Represented by the symbol + or v. It works on more than one proposition and the output is true when any of the proposition is true. Disjunction means either of the proposition is true.

Conditional (if ... then or Implication) : Represented by the symbol  or  or . Implication means if one proposition is true then other proposition is true.

Bi-conditional (if and only if or Equivalence) : Represented by the symbol or  . Equivalence means both propositions are true or both are false.

Truth Table and Truth Values : A Truth Table is a complete list of possible truth values of a proposition and the Truth Values are defined as truth or falsity of a proposition.

Truth Tables of all the connectives : 

Related Terms

Contingencies : The proposition that have some combination of 1’s and 0’s in their truth table column, are called Contingencies.

Tautology : The proposition that have all 1’s in their truth table column, are called Tautology.

Contradictions : The proposition that have all 0’s in their truth table column, are calledContradictions.

Consistent Statement : Two statements are consistent if and only if their conjunction is not a contradiction.

Converse : The converse of a conditional proposition is determined by interchanging the antecedent (The first proposition) and consequent (The second proposition) of given conditional.  if  X => Y then its converse is Y => X.

Inverse : The inverse of a conditional proposition is another conditional having negated antecedent and consequent.  if  X => Y then its inverse is X' => Y'.

Contrapositive : The contrapositive of a conditional is formed by creating another conditional that takes its antecedent as negated consequent of earlier condition and consequent as negated antecedent of earlier conditional. If  X => Y then its contrapositive is Y' => X'

Gaurav Nigam

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Boolean simplification

Simplification of Boolean Expressions

Here are some examples of Boolean algebra simplifications. Each line gives a form of the expression, and the rule or rules used to derive it from the previous one. Generally, there are several ways to reach the result. Here are some of the examples for you as guideline :

• Simplify: AB(A + B)(B + B):

Expression	Rule(s) Used
AB(A + B)(B + B)	Original Expression
AB(A + B)	Complement law, Identity law.
(A + B)(A + B)	DeMorgan's Law
A + BB	Distributive law. This step uses the fact that or distributes over and. It can look a bit strange since addition does not distribute over multiplication.
A	Complement, Identity.

• Simplify: (A + C)(AD + AD) + AC + C:

Expression	Rule(s) Used
(A + C)(AD + AD) + AC + C	Original Expression
(A + C)A(D + D) + AC + C	Distributive.
(A + C)A + AC + C	Complement, Identity.
A((A + C) + C) + C	Commutative, Distributive.
A(A + C) + C	Associative, Idempotent.
AA + AC + C	Distributive.
A + (A + T)C	Idempotent, Identity, Distributive.
A + C	Identity, twice.

• Simplify: A(A + B) + (B + AA)(A + B):

Expression	Rule(s) Used
A(A + B) + (B + AA)(A + B)	Original Expression
AA + AB + (B + A)A + (B + A)B	Idempotent (AA to A), then Distributive, used twice.
AB + (B + A)A + (B + A)B	Complement, then Identity. (Strictly speaking, we also used the Commutative Law for each of these applications.)
AB + BA + AA + BB + AB	Distributive, two places.
AB + BA + A + AB	Idempotent (for the A's), then Complement and Identity to remove BB.
AB + AB + AT + AB	Commutative, Identity; setting up for the next step.
AB + A(B + T + B)	Distributive.
AB + A	Identity, twice (depending how you count it).
A + AB	Commutative.
(A + A)(A + B)	Distributive.
A + B	Complement, Identity.

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