This unit provides the foundations for much of University level mathematics.  It introduces rigorous mathematical language and notation that are used throughout the degree, including mathematical logic.  In it we discuss methods of proof, properties of finite and infinite sets as well as equivalence relations. In the second half, we introduce the notion of convergence of infinite sequences and their limits and finally make rigorous the idea of continuous function. There are also discussions of complex numbers.

The unit aims to introduce students to the foundations of pure mathematics, including methods of proof, mathematical logic, sets, functions and mathematical analysis.

On the successful completion of the course, students will be able to:

• Analyse the meaning of mathematical statements involving quantifiers and logical connectives and construct the negation of a given statement.
• Construct elementary proofs of mathematical statements using a range of fundamental proof techniques (direct argument, induction, contradiction, use of contrapositive).
• Use basic set theoretic language and constructions, determine the cardinality of a set and recognise equivalence relations on sets.
• Perform calculations with complex numbers in standard and exponential form.
• Verify properties of functions and construct composite functions and inverse functions.
• Determine convergence and limits of sequences directly from definitions and using properties of convergent sequences.
• Construct proofs and counter-examples about the convergence of sequences of real and complex numbers.
• Determine whether a real-valued function is continuous at a point and calculate the limit of a function at a point.
• Construct proofs and counter-examples about continuity of functions for real-valued functions defined on subsets of the reals.

There is a supervision each week which provides an opportunity for students' work to be marked and discussed and to provide feedback on their understanding. Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

Peter Eccles ‘ An Introduction to Mathematical Reasoning’ , CUP, 1997
Mary Hart ‘Guide to Analysis’, Red Globe Press, 2001
K G Binmore ‘Mathematical Analysis: a straightforward approach’, CUP, 1982
Kevin Houston, ‘How to Think Like a Mathematician’ CUP, 2009
Lara Alcock ‘How to think about Analysis’, OUP, 2014.