Mathematical Logic

Statement A statement is defined as a declarative sentence, which is either true or false, but not both at a time. It is also known as a proposition or an atomic statement. Examples: Consider the following sentences,  “The earth rotates around the sun” This sentence is declarative as it is giving us some information. It is a true statement. Hence it is a logical statement or a proposition.  “It is not raining today. “ This sentence is also giving some information. It is either true or false, but of course, not both at any particular time. Hence it is a Statement. Truth-value of a statement: The truth-ness or falsity of the statement is known as the truth-value of the statement. If the statement is true, we say that its truth-value is true and denote it by the letter ‘T’ or by the number “1”. If the statement is false, we say that its truth-value is False and it is denoted by the letter ‘F’ or by the number ‘0’. Examples: Consider the following sentences,  “The earth rotates around the sun”. This statement is true. Hence it is a logical Statement, which has truth-value “T”.  “The positive divisors of 10 are 1, 2, 5 and 10 only. “ This statement is true. So it has truth-value “T”.  “4 + 15 > 20.” This statement is false. So it has truth-value “F”. Logical connectives In our day-to-day language, we form new sentences with two or more sentences. Similarly in mathematical logic we use some words to form new statements. Such words are called logical connectives. Types of logical statements Simple statement: A logical statement in which no connective is used is called a “simple statement “or a proposition or an atomic statement . Generally small case letters are use to denote simple statement. Examples: p: London is in England. q: 2 + 7 = 10. r: 21 is a multiple of 9. s: 11  121 Compound statement: If a statement is formed by joining 2 or more simple statements using logical connectives then it is called a “compound statement” or “a propositional form. Examples: p: Mumbai is in England and it is raining. q: 2 + 7 = 10 or 1 is a prime number. r: If 21 is a multiple of 9 then 2+ 2= 4. s: It is not true that , 11  121. 1. Negation: Negation is the compound statement in which the word “not" is the connective used. If ‘p’ is a simple statement, then the statement ‘not p’ is known as the negation of p. It is denoted symbolically as ‘~p’. 2. Conjunction is the compound statement in which connective used is the word “and”. Suppose ‘p’ and ‘q’ are any two statements. The conjunction of ‘p’ and ‘q’ is defined as the statement ‘p and q’. It is denoted symbolically as p^q. Example: If p : Sunday is a holiday, q : Every day I study for at least 4 hours, 3. Disjunction is the compound statement in which the connective “or” is used. Suppose p and q are two statements then the disjunction of ‘p’ and ‘q’ is the statement ‘p or q’. It is denoted symbolically as pVq. 4. Conditional Statement or Implication: It is a compound statement in which, we combine two statements by the phrase “If – then” or “implies that”. This gives us a new statement. Suppose p and q are two statements then the compound statement “If p, then q” is called a conditional statement. We denote this conditional statement symbolically as p --> q. 5. Biconditional Statement or Double Implication: It is a compound statement in which, we combine two statements by the phrase “If and only if”. Suppose p and q are any two statements, then the compound statement “p if and only if q” is called a biconditional statement. It is denoted by p <--> q