1. Joan Ridgway

2. If a proposition is not indeterminate then it is either true (T) or false (F). True and False are complementary events. For two propositions, they are either both true or both false, or one is true and one is false: TT, FF, TF, or FT (which reminds us of the situation when two coins are tossed) A truth table is a good way to show the possibilities For two propositions p, q For three propositions p, q, r pq TT TF FT FF pqr TTT TTF TFT TFF FTT FTF FFT FFF To fill in the truth tables for complicated compound propositions, we build them up in simple stages, filling in the columns step by step.

3. pq  q TT TF FT FF pq TTF TF FT FF pq TTF TFT FT FF pq TTF TFT FT F FF pq TTF TFT FT F FFT NOT (Negation)

4. pq  q TTF TFT FT F FFT NOT (Negation)AND (Conjunction) pq p  q TT TF FT FF pq TTT TF FT FF pq TTT TFF FT FF pq TTT TFF FTF FF pq TTT TFF FTF FFF OR (Disjunction) pq p  q TT TF FT FF pq TTT TF FT FF pq TTT TFT FT FF pq TTT TFT FTT FF pq TTT TFT FTT FFF

5. pq  q TTF TFT FT F FFT NOT (Negation)AND (Conjunction) pq p  q TTT TFF FTF FFF OR (Disjunction) pq p  q TTT TFT FTT FFF OR BUT NOT BOTH (Exclusive disjunction) pq p  q TT TF FT FF pq TTF TF FT FF pq TTF TFT FTT FFF pq TTF TFT FT FF pq TTF TFT FTT FF

6. pq TTF TFT FTT FFF pq  q TTF TFT FT F FFT NOT (Negation)AND (Conjunction) pq p  q TTT TFF FTF FFF OR (Disjunction) pq p  q TTT TFT FTT FFF OR BUT NOT BOTH (Exclusive disjunction)

7. IF... THEN... Implication pq p  q TT TF FT FF pq TTT TF FT FF pq TTT TFF FT FF pq TTT TFF FTT FF pq TTT TFF FTT FFT

8. pq TTT TFF FTT FFT IF... THEN... Implication This is the one which is the least intuitive! An example might help. p : You get into university. q : You parents buy you a car. Your parents say “If you get into university, then we will buy you a car.” Suppose your parents have made you a promise. There are four routes that the events might take: 2) You get into university, and your parents do not buy you a car. Then p  q is false. 1) You get into university, and your parents buy you a car. Then p  q is true. 3) You do not get into university, but your parents buy you a car anyway. Then p  q is still true. You just have very generous parents! 4) You do not get into university, and your parents do not buy you a car. Then p  q is true. They did not break their promise!

9. Show that the truth table values for the two propositions below are identical.  p  q and p  q pq TT TF FT FF Some mathematicians like to simply define implication by saying that p implies q is the same as the proposition  p  q. That is: p  q =  p  q p : You send me the money first. q : I will send you the CD. “If you send me the money first, then I will send you the CD” is logically equivalent to “Either you don’t send me the money or I will send you the CD”

10. Show that the truth table values for the two propositions below are identical.  p  q and p  q pq  p p TT TF FT FF  p  q p  q T F T T T F T T p  qp  q Some mathematicians like to simply define implication by saying that p implies q is the same as the proposition  p  q. That is: p  q =  p  q p : You send me the money first. q : I will send you the CD. “If you send me the money first, then I will send you the CD” is logically equivalent to “Either you don’t send me the money or I will send you the CD” So  p  q and p  q are logically equivalent (identical). F F T T

11. A tautology is a proposition which is always true, no matter what the truth values of its constituent parts are. Tautology In a truth table, a tautology will end up with a T in every row. An obvious example of a tautology is the proposition p   p Either p is true or it is not true. p  p p p   pp   p T F F T T T

12. A logical contradiction is a proposition which is always false, no matter what the truth values of its constituent parts are. In a truth table, a logical contradiction will end up with an F in every row. An obvious example of a logical contradiction is the proposition p   p p cannot be both true and not true at the same time. p  p p p   pp   p T F F T F F Logical Contradiction

13. The converse of p  q is q  p. Only an implication can have a converse. Converse Just because an implication is true, it does not follow that the converse must be true (but it sometimes is). For example, the converse of “If it was sunny, then I walked to work” is “If I walked to work, then it was sunny”. The converse of “All Frenchmen are Europeans are” is “All Europeans are Frenchmen”. The converse of “If x is odd then x 2 is odd” is “if x 2 is odd, then x is odd”.

14. The inverse of p  q is  p   q. Only an implication can have a inverse. Inverse Just because an implication is true, it does not follow that the inverse must be true (but it sometimes is). For example, the inverse of “If it was sunny, then I walked to work” is “If it was not sunny then I did not walk to work”. The inverse of “All Frenchmen are Europeans are” is “If someone is not French then they are not a European”. The inverse of “If x is odd then x 2 is odd” is “if x is even, then x 2 is even”.

15. The contrapositive of p  q is  q   p. Only an implication can have a contrapositive. Contrapositive If an implication is true, the contrapositive must be true. For example, the contapositive of “If it was sunny, then I walked to work” is “If I did not walk to work then it was not sunny”. The contrapositive of “All Frenchmen are Europeans are” is “If someone is not European then they are not French”. Thecontrapositive of “If x is odd then x 2 is odd” is “if x 2 is even, then x is even”.

16. T F F F T T F F T T T T All true, so a tautology.  q  p q  p

17. F T F T F T T T F F T T T T T F This is not a tautology, because the column for ( p   q ) is not identical to the column for (  p  q ) ( p   q ) is not logically equivalent to (  p  q )