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CHAPTER Some Mathematical Topics on Which to Practice Proof Techniques 4 The information presented in the previous chapters gives the nuts and bolts of the most common and basic techniques used to prove the truth of mathematical results. These techniques can be implemented in all branches of mathematics. The present chapter introduces the elementary concepts of some of the most commonly studied branches of mathematics in the undergraduate curriculum, but no new proof techniques (strictly speaking). The material presented will show how the basic proof techniques from the previous chapters can be put to good use. BASIC SET THEORY AND INDEXED FAMILIES A set is a well-defined collection of objects. The objects that belong to a set are called the elements of the set. If x is an element of a set A we write x A. The empty set is a set with no elements, usually repre- sented either by or by { }.