Discrete Math

Project #1 Due Friday Spring 2020 Math2225 1. Give an example of a compound proposition with three variables that is a contradiction. Justify your answer with a truth table. 2. For each implication, state the (i) converse and (ii) contrapositive. (a) If r is positive then the sun is shining. (b) Sadie goes to the vet whenever 5 is prime. (c) Cats having horns is sufficient for unicorns to exist. 3. One hundred statements are written down in order. The nth statement in the list reads "Exactly n of the statements in this list are false." Determine the truth values for each of the statements. Be sure to explain your answer. 4. One hundred statements are written down in order. The nth statement in the list reads "At least n of the statements in this list are false." Determine the truth values for each of the statements. Be sure to explain your answer. 5. There is a special island in the South Pacific on which there are three types of animals who can talk: kangaroos who always tell the truth, hyenas who always lie, and sloths who can either lie or tell the truth. You come across three animals, A, B, and C, but cannot tell what type of animal they are as you have been stricken blind by a rare virus. Each of the three people knows the type of person each of other two is. For each of the following situations (if possible determine whether there is a unique solution and determine who the kangaroo, hyena, and sloth are. When there is no unique solution, list all possible solutions or state that there are no solutions. (a) A says "C is the hyena" B says "A is the kangaroo" C says "I am the sloth" (b) (27) A says "I am the kangaroo" B says "A is telling the truth" Csays "I am the sloth" (c) A says "I am not the sloth" B says "I am not the sloth" C says "I am the sloth" (d) A says "I am the hyena" B says "A is the hyena" C says "I am the hyena" Project #1 Due Friday Spring 2020 Math2225 1. Give an example of a compound proposition with three variables that is a contradiction. Justify your answer with a truth table. 2. For each implication, state the (i) converse and (ii) contrapositive. (a) If r is positive then the sun is shining. (b) Sadie goes to the vet whenever 5 is prime. (c) Cats having horns is sufficient for unicorns to exist. 3. One hundred statements are written down in order. The nth statement in the list reads "Exactly n of the statements in this list are false." Determine the truth values for each of the statements. Be sure to explain your answer. 4. One hundred statements are written down in order. The nth statement in the list reads "At least n of the statements in this list are false." Determine the truth values for each of the statements. Be sure to explain your answer. 5. There is a special island in the South Pacific on which there are three types of animals who can talk: kangaroos who always tell the truth, hyenas who always lie, and sloths who can either lie or tell the truth. You come across three animals, A, B, and C, but cannot tell what type of animal they are as you have been stricken blind by a rare virus. Each of the three people knows the type of person each of other two is. For each of the following situations (if possible determine whether there is a unique solution and determine who the kangaroo, hyena, and sloth are. When there is no unique solution, list all possible solutions or state that there are no solutions. (a) A says "C is the hyena" B says "A is the kangaroo" C says "I am the sloth" (b) (27) A says "I am the kangaroo" B says "A is telling the truth" Csays "I am the sloth" (c) A says "I am not the sloth" B says "I am not the sloth" C says "I am the sloth" (d) A says "I am the hyena" B says "A is the hyena" C says "I am the hyena