Math 55 Worksheet June 22, 2011Adapted from worksheets by Rob Bayer, Summer 2009.Instructions• Introduce yourselves! Despite popular belief, math is in fact a team sport!• Decide who will be your first scribe.• Do the problems below, having a different person be the scribe for each one.• Try to work out the problems as a group, but feel free to flag me down if you run into a wall.Rules of inference1. Determine what conclusions, if any, you can draw from each of the following sets of premises:(a) “All athletes watch professional sports,” “Alice is an athelete.”(b) “All athletes watch professional sports,” “Bob watches professional sports.”(c) “All athletes watch professional sports,” “Everyone who watches sports likes pizza,” “Charlie doesn’t likepizza.”(d) “All athletes watch sports,” “Dan isn’t an athelete.”2. What’s wrong with each of the following arguments?(a) Every CS major takes discrete math. Alice is taking discrete math. Therefore, Alice is a CS major.(b) Every CS major takes discrete math. Bob is not a CS major. Therefore, Bob doesn’t take discrete math.(c) If n > 1, then n2> 1. Since (−5)2= 25 > 1, it follows that−5 > 1.3. Beginning with the premises “no pigs can fly,” and “some birds can fly,” use rules of inference to conclude that“some birds are not pigs.”Methods of proof1. Prove that if 3n + 2 is odd, then n is odd.2. Prove that, for a positive integer n, n is even if and only if 7n + 4 is even.3. What’s wrong with the following “proof” that if m2= n2, then m = n?“We want to show m = n. Squaring both sides gives m2= n2, which is true by hypothesis. Therefore, we musthave m = n.”4. Prove that the sum of a rational number and an irrational number is irrational.5. Prove that if a, b, n are positive integers such that ab > n, then a >√n or b >√n.Logic Puzzles1. F While walking through the woods on the Island of Knights and Knaves you come to a fork in the trail. Onepath leads to a deep dark pit of doom and the other leads to a boat that can take you home. There are no signsindicating which direction to take, but there is one inhabitant of the island there waiting for you. Given thatyou don’t know if he’s a Knight or a Knave, what one question would you ask him to determine the correctpath? HINT: This question need not determine whether he is a Knight or a Knave.2. Consider the following set of four statements:(a) One of these statements is false(b) Two of these are false(c) Three of these are false(d) Four of these are falseWhich of the above, if any, are true?3. Two bicyclists enter opposite ends of a 100-foot long tunnel that is only wide enough for one bike. One istravelling 10 ft/s and the other is travelling 5 ft/s in the opposite direction. A bird flying 20 ft/s enters thetunnel just in front of the 10 ft/s cyclist. When the bird gets to the other cyclist, it immediately turns aroundand flies back towards the 10 ft/s cyclist. When the bird gets back to him, it turns around again, etc, etc. Howmany feet does the bird fly before the bicyclists collide?4. F Four people travelling at night come to a small footbridge and need to cross to the other side. Unfortunately,they only have 1 flashlight and the bridge can only hold the weight of two people at once. One person takes 10minutes to cross, another 5, another 2, and the last takes 1 minute. Anyone crossing must have the flashlightand when travelling together, they must go the pace of the slower person. What is the quickest time in whichthat can all get