Chapter 1 Section 1.4 More on Conditionals

There are three statements that are related to a conditional statement. They are called the converse, inverse and contrapositive statements. Converse Interchanges the hypothesis and conclusion: q → p Original Statement: If you pay your electric bill on time then the power company will keep your electricity on. Converse Statement: If the power company keeps your electricity on then you pay your electric bill on time. pq q → p TTT TFT FTF FFT Inverse Negates the hypothesis and conclusion: (~ p ) → (~ q ) Original Statement: If you pay your electric bill on time then the power company will keep your electricity on. Inverse Statement: If you do not pay your electric bill on time then the power company will not keep your electricity on. Contrapositive Negates and Interchanges the hypothesis and conclusion : (~ q ) → (~ p ) Original Statement: If you pay your electric bill on time then the power company will keep your electricity on. Contrapositive Statement: If the power company is not keeping your electricity on then you do not pay your electric bill on time. pq ~p~p ~q~q (~ p ) → (~ q ) TTFFT TFFTT FTTFF FFTTT pq ~p~p ~q~q (~ q ) → (~ p ) TTFFT TFFTF FTTFT FFTTT

The Biconditional Statement This is a statement that acts like an equal sign for logical statements. It is true when the two statements have the same truth value. It is often formed with the “if and only if” sentence construction. Uses the symbol p ↔ q. You pay you electric bill on time if and only if the power company keeps your electricity on. pq p ↔ q TTT TFF FTF FFT pq p → qq → p ( p → q ) Λ ( q → p ) TTTTT TFFTF FTTFF FFTTT The truth table to the right shows that the biconditional statement is logically equivalent to the conjunction of the statement and its converse. Notice the last columns of each of the truth tables above are the same which means the statements p ↔ q and ( p → q ) Λ ( q → p ) are logically equivalent. We use the symbol ≡ like and equal sign for numbers. We can "replace" the statement ( p → q ) Λ ( q → p ) by p ↔ q : ( p → q ) Λ ( q → p ) ≡ p ↔ q

Consider the truth table for each of the examples below. I. Truth Table for: p V (~ p )II. Truth Table for: p Λ ( q Λ (~ q )) p ~p~pp V (~ p ) T T T TF F Notice that the statement above will always evaluate to be true no matter what initial values of p we start with (look at the last column). This type of statement that evaluates to be true (T) no matter what values we start with is called a tautology. Notice: p V (~ p ) ≡ T pq ~q~qq Λ (~ q ) p Λ ( q Λ (~ q )) T F T T T T T F F F F F F F F F F F F F Notice that the statement above will always evaluate to false no matter what the initial values for p and q you start with (look at the last column). This type of statement evaluates to be false (F) no matter what values we start with is called a contradiction. Notice: p Λ ( q Λ (~ q )) ≡ F

Law of Double Negation The statements p and the ~(~ p ) are logically equivalent. We can see this by looking at the truth tables below: pp TT FF p ~p~p ~(~ p )) TFT FTF We can "replace" p by ~(~ p ) or in terms of logic: p ≡ ~(~ p ) Law of Contraposition The statements p  q and ~ q  ~ p are also logically equivalent. pq p  qp  q TTT TFF FTT FFT pq ~p~p ~q~q ~ q  ~ p TTFFT TFFTF FTTFT FFTTT We can "replace" p  q by ~ q  ~ p or in terms of logic: p  q ≡ ~ q  ~ p

DeMorgan's Laws DeMorgan's Laws are ways to "distribute" a negation inside parenthesis where the statements inside are being connected by an "and" (Λ) or and "or" (V). pq p Λ qp Λ q ~( p Λ q ) TTTF TFFT FTFT FFFT pq ~p~p ~q~q (~ p ) V (~ q ) TTFFF TFFTT FTTFT FFTTT This says that: ~( p Λ q ) ≡ (~ p ) V (~ q ) It is also true that: ~( p V q ) ≡ (~ p ) Λ (~ q ) Conditionals The implication statement p  q is equivalent to the statement (~ p ) V q. Which says that p  q  (~ p ) V q. Rewrite the conditional statement "If you can read then you can learn". Let p : You can read. q : You can learn. p  q : If you can read then you can learn. (~ p ) V q : You can not read or you can learn.