What do you understand by 'Logic' and 'Propositional Logic'? Logic is a collection of rules for reasoning. In logic, we discuss about true or false of the statements and how to determine it with the help of other statements.A proposition is a sentence that is either true or false. Propositional Logic is the logic of sentences.
What is a proposition? A proposition is a sentence that is either true or false.
How will you distinguish a proposition from a compound proposition? A proposition represented as a simple sentence is called simple proposition whereas when two or more propositions are joined together with the help of some connecting words then the resulting proposition is said to be 'Compound Proposition'.
Define the following terms with a suitable example for each of them: (a) Conjunction Conjunction joins two or more propositional statements with the help of 'and' connective. It is represented by symbol '^' or '.'(dot). The conjunction of two or more propositions results in true if all the propositions are true otherwise, false.For example: a: Sky is clear. b: You can go out. The conjunction of above mentioned propositions will be 'Sky is clear and you can go out'. It can be represented in expression form as (a ^ b). Alternatively, it can also be written as (a.b). (b) Disjunction Disjunction is the term used to combine two or more propositional statements with 'or' connective. It is represented with the symbol 'v' or '+'. It results in false if both the operands are false otherwise, true.For example: a: 8 is divisible by 2. b: It is an even number. These two propositions are combined using Disjunction like this — '8 is divisible by 2 or it is an even number.' It can be represented in expression form as (a v b). Alternatively, it can also be written as (a + b). c) Converse A conditional obtained by interchanging the antecedent and consequent of the given conditional is known as Converse. The Converse of a given conditional 'If a then b' will be written as 'If b then a'.For example: If 16 is divisible by 2 then it is an even number. Converse --If 16 is an even number then it is divisible by 2. (d) Inverse Inverse is a conditional statement that can be obtained by negating the antecedent and consequent of the given conditional. The inverse of a given conditional 'If a then b' can be written as 'If ~a then ~b'.For example: If you practice well then you will win the match. Inverse— If you don't practice well then you won't win the match.
Write converse, inverse and contrapositive of each statement: (a) If it rains, then you will not play.
Converse— If you will not play then it will rain. Inverse— If it doesn't rain then you will play. Contrapositive— If you will play then it will not rain.
(b) If you work hard, then you will pass.
Converse— If you will pass then you worked hard. Inverse— If you don't work hard, then you will not pass. Contrapositive— If you will not pass then you didn't work hard.
(c) If I run fast, then I will win the race.
Converse— If I will win the race then I ran fast. Inverse— If I don't run fast, then I will not win the race. Contrapositive— If I will not win the race then I didn't run fast.
Differentiate between proposition and wff. A proposition is a sentence that is either true or false whereas wff (Well-Formed Formula) is a system of representing a propositional statement or expression in short form.
Define the term Contingency, Contradiction and Tautology. Contingency When a propositional statement is concluded into true as well as false for different values of the variables, it is said to be Contingency. Contradiction When a propositional statement is false for all values of the variables, it is said to be Contradiction. Tautology When a propositional statement is true for all values of the variables, it is said to be Tautology.
What are the fundamental concepts of boolean algebra? The fundamental concept of boolean algebra is to deal with logical problems in mathematics by using only two values i.e. digits 0 (zero) and 1 (one) or 'False' and 'True' or 'ON' and 'OFF' logical states.
What is truth table? Explain with reference to boolean algebra. Truth Table is the tabular representation of the values given and the result obtained due to any logical operation.
What do you mean by binary valued quantities? The digits 0 (zero) and 1 (one) of the binary number system used in boolean algebra are called binary valued quantities.
What do you understand by Karnaugh's map? Explain. Karnaugh's map is a way to reduce an expression by using a tabular or matrix representation to its most simplified form. It is named after its inventor Maurice Karnaugh. Its advantage is that it doesn't require any algebraic derivation. It has a limitation that it will reduce the expression only when it is in canonical SOP or POS form.