The Foundations: Logic and Proofs



  • Propositions : A proposition is a declarative sentence (that is, a sentence that declares a fact) that is either true or false, but not both.


    1. Washington, D.C., is the capital of the United States of America.


    2. Toronto is the capital of Canada.


    3. 1 + 1 = 2.


  • Propositions 1 and 3 are true, whereas 2 and 4 are false.


  • The truth value of a proposition is true, denoted by T, if it is a true proposition, and the truth value of a proposition is false, denoted by F, if it is a false proposition.


  • The area of logic that deals with propositions is called the propositional calculus or propositional logic.


  • New propositions, called compound propositions, are formed from existing propositions using logical operators.


  • Let p be a proposition. The negation of p, denoted by ¬p (also denoted by p'), is the statement. The proposition ¬p is read “not p.” The truth value of the negation of p, ¬p, is the opposite of the truth value of p.



Truth Table for Negation P


    P¬P
    0 1
    1 0




Connectives - Conjunction




Truth Table for Conjunction : p ∧ q






Connectives - Disjunction




Truth Table for Disjunction : p ∨ q






Connectives - Exclusive OR




The Truth Table for the Exclusive Or of Two Propositions : p ⊕ q




Conditional Statements







CONVERSE, CONTRAPOSITIVE, AND INVERSE




Q : What are the contrapositive, the converse, and the inverse of the conditional statement “The home team wins whenever it is raining?”






BICONDITIONALS






Q. Let p be the statement “You can take the flight,” and let q be the statement “You buy a ticket.” Then p ↔ q is the statement






Truth Tables of Compound Propositions



Q. Construct the truth table of the compound proposition (p ∨ ¬q) → (p ∧ q).






Precedence of Logical Operators








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