Assign a grade of A (correct), C (partially correct), or F (failure) to each. Justifyassignments of grades other than A. (a) Claim.Proof.(b) Claim. If and then .Proof. Suppose Then there existswith . But implies that andwhereas implies that and However,and so and This is a contradiction. Therefore,(c) Claim. If and thenProof. Suppose ThenTherefore (d) Claim. If and thenProof. To show suppose Choose anyThen But since ThusThis proves A proof of is similar. Therefore,(e) Claim. Let R and S be relations from A to B and from B to C, respectively.ThenProof. The pairTherefore,(f) Claim. Let R be a relation from A to B. ThenProof. Suppose Choose any such thatThen, Thus Therefore,(g) Claim. Suppose R is a relation from A to B. ThenProof. Let Then for some andThus Since and Thus(x, y) = (x, x) and so x A, (x, y) IA.(z, y)

Introduction to Mathematical Thinking Learn how to think the way mathematicians do a powerful cognitive process developed over thousands of years. About the Course NOTE: Coursera encountered difficulties in converting my course to run on the new platform. Working together, we have found a way to modify the course to circumvent the missing platform features, without losing too much of what made the course work. Completing that work will involve considerable time and effort, and I am unlikely to have much time to look at this until the summer. This means that the earliest Session 8 could run is Fall 2016. Please check back here in August. Sorry about this. Keith Devlin, 1/25/2016 (modified 4.21.2016) The goal of the course is to help you develop a valuable mental ability – a