1 B Truth Tables and Equivalent Propositions Example
§ 1. B Truth Tables and Equivalent Propositions
Example of a Truth Table for a Complex Proposition Consider the compound proposition: “The Nationals win the pennant and either Joe catches a fly ball or Joe makes two errors. ” We let p be the statement “The Nationals win the pennant. ” We let q be the statement “Joe catches a fly ball” Finally, let r be the statement “Joe makes two errors. ” The compound statement is thus p˄(q˅r).
Example of a Truth Table for a Complex Proposition (Continued) We construct the following truth table: p T T T r T T F q T F T q˅r T T T p˄(q˅r) T T F F F T F F T T T F F F
Example of a Truth Table for a Complex Proposition (Continued) You will note that we added a partial results column headed “q˅r” to the truth table to get an intermediate result that simplifies the finding of the truth of the compound proposition p˄(q˅r). When we have a compound proposition that has significant logic within the proposition the technique of adding columns for intermediate results is extremely helpful in finding the overall truth value of the compound proposition.
Example of Truth Values for Mathematical Propositions Let p be the proposition “ 6 = 4 + 2” Let q be the proposition “ 5 < 1” Let r be the proposition “ 2 + x < 2” Evaluate the truth value of the compound proposition p˄~(q˅r). Here, p has a value T always and q has a value F always. But r can have either a value of T or F dependent on the value of the variable x. So we can construct the following truth table: p T T q F F r T F q˅r T F ~(q˅r) F T p˄ ~(q˅r) F T
Equivalent Propositions and De Morgan’s Laws Two propositions are equivalent if they have the same truth values in every possible situation. Examples: Ø “ 6 = 4+2” is equivalent to “ 2 < 35” since they are both true. Ø p˄q is equivalent to ~(~p˅~q) (Exercise: Check the truth tables. ) Ø “The name “John” starts with a “W”. ” is equivalent to “ 5 < 1”. For any two statements p and q, the following is always true: Ø ~(p ˅ q) is equivalent to ~p ˄ ~q Ø ~(p ˄ q) is equivalent to ~p ˅ ~q These two equivalencies are known as De Morgan’s Laws.
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