By Martin Aigner

This e-book bargains a well-organized, easy-to-follow creation to combinatorial concept, with examples, notes and workouts. ". . . a great advent to combinatorics. This e-book can warmly be steered to begin with to scholars drawn to combinatorics." Publicationes Mathematicae Debrecen

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In other words, the statements "P,* Q" and "not Q both True for the same logical values of P and Q. 2 1. The implication If the sky is not blue then this is not earth has as contrapositive If this is earth then the sky is blue. The implication and its contrapositive are making the same logical statement about the sky. 2. The implication If my CPS is working then I am not lost has contrapositive If I am lost then my CPS is not working. Notice that both the implication and its contrapositive are making the same logical statement, assuming I always use my GPS.

Use a counterexample and a proof by contradiction. 8. Prove that No opinion is valid is False. Use a proof by contradiction. 9. This is a hard one. Find the logical state of I am lying when it exists outside of the Epimenides Paradox. 10. Refer to # 9. ruth Sayer then can you speak I am lying? 11. Let P be a statement that has an unnamed logical state S. Does P have logical state S in every conversation that contains it? Chapter 2 Sets In this book, we will count the number of elements or objects that have a certain given property.

There are 23 such elements. Thus P( {a, b, c}) = {0, {a}, {b}, {c}, {a,b},{a,c},{b,c},{a,b,c}}. 6. Write down P( {a, b, c, d}). 7. Let A = {2, 3, 4, 5}, B = {2, 3, 4, 5, 6}, C = {1, 2, 6} and let U = {O, 1,2,3,4,5,6,7,8, 9}. Find and compare these sets. 1. An(BUC)and(AnB)U(AnC). Answer: An(BUC)= {2, 3, 4, 5} n {1, 2, 3, 4,5, 6} = {2, 3, 4, 5}. 2. A' n B' and (A U B)'. Answer: A' n B' = {O, 1, 7, 8, 9}. (A U B)' = {O, 1,7,8, 9}. Notice that A' n B' = (A U B)'. 3. A' U B' and (A n B)'. 8. Show that the following are equal.