Homework #1 - Discrete Mathematics[1].doc

Sean E. CorpuzDiscrete MathematicsJanuary 13, 2004Dr. Hairong KuangHomework #15. Several forms of negation are given for each of the following statements. Which are correct?a. The answer is either 2 or 3.1. Neither 2 nor 3 is the answer.2. The answer is not 2 or not 3.3. The answer is not 2 and it is not 3.Answer: In this particular situation there are two forms which negate the statement above. They are: 1 and 3. b. Cucumbers are green and seedy.1. Cucumbers are not green and not seedy.2. Cucumbers are not green or not seedy.3. Cucumbers are green and not seedy,Answer: In this case, 1, 2, and 3 all negate the statement cucumbers are green AND seedy.c. 2 7 and 3 is even.2. 2 7 and 3 is even.3. 2 7 or 3 is odd.4. 2 7 or 3 is even.Answer: Finally statements 1, 2, 3 and 4 all negate the primary statement above.13. Using letters for component statements, translate the following compound statements into symbolic notation.a. If Anita wins the election, then tax rates will be reduced.Answer:A: Anita wins the election.B: Tax rates will be reduced.A Bb. Tax rates will be reduced only if Anita wins the election and the economy remains strong.Answer:A: Tax rates will be reduced.B: Anita wins the election.C: The economy remains strong.(B C) Ac. Tax rates will be reduced, if the economy remains strong.Answer:A: Tax rates will be reduced.B: The economy remains strong.B Ad. A strong economy will follow from Anita winning the election.Answer:A: A strong economy will follow.B: Anita winning the election.B Ae. The economy will remain strong if and only if Anita wins the election or tax rates are reduced.Answer:A: The economy will remain strong.B: Anita wins the election.C: Tax rates are reduced.(B / C) A14. Construct truth tables for the following well formed formulae. Note any tautologies or contradictions.c. A / (A / B)Answer:ABAB(A / B)A / (A / B)0011 1 00110 1 01001 1 11100 0 0This well formed formula contains no tautology or contradiction whatsoever. e. (A B) (A / C) (B / C) Answer:A B C (A B) (A / C) (B / C) (A B) (A / C) (B / C)0 0 0 10 0 10 0 1 11 1 10 1 0 10 1 10 1 1 11 1 1 1 0 0 01 0 1 1 0 1 01 1 1 1 1 0 11 1 1 1 1 1 11 1 1Please take note that this particular well formed formula is a tautology. g. A / B B / AAnswer:ABABA / BB / A A / B B / A0011 0 1 00110 0 1 01001 0 1 01100 0 0 0Please take note that this well formed formula is a contradiction with any value.i. (A / B) / C A / CAnswer:ABCABC(A / B)A / C(A / B) / C 000111 0 1 0001110 0 1 0010101 0 1 0011100 0 1 0100011 0 0 0101010 0 0 0110001 1 0 1111000 1 1 0(A / B) / C A / C1111110117.Verify each with a truth table:d. A A / BAnswer:ABA / BA A / B00 0 101 1 110 1 111 1 1Please note that this is a tautology.f. (A / B) A / BABAB(A / B)A / B(A / B) A / B0011 1 110110 1 111001 1 111100 0 01Note that this well formed formula is a tautology.Of all the statements which were given to us, only four are propositions. They are the following:c. There are no black flies in Maine.In this case, the truth value of this statement lies within the fact that there are no black flies in Maine.d. 4 + x = 5The truth value for this particular statement is x = 1.e. x + 1 = 5 if x =1This statement with the given values is false; however, it doesnt negate the fact that this is