Sets for Mathematics
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Advanced undergraduate or beginning graduate students need a unified foundation for their study of geometry, analysis, and algebra. For the first time, this book uses categorical algebra to build such a foundation, starting from intuitive descriptions of mathematically and physically common phenomena and advancing to a precise specification of the nature of Categories of Sets. Set theory as the algebra of mappings is introduced and developed as a unifying basis for advanced mathematical subjects such as algebra, geometry, analysis, and combinatorics. The formal study evolves from general axioms that express universal properties of sums, products, mapping sets, and natural number recursion.
from 0) is called an initial object of the category. 2.6 Additional Exercises 45 (c) Show that the zero vector space is simultaneously a terminal object and an initial object (such an object is called a zero object). (d) Show that any part of a vector space has an inverse image along any linear transformation. Remember to verify that the inverse image of the mapping is a vector space. Exercise 2.41 (a) Show that injective group homomorphisms are monomorphisms in the category of groups and
will be established later in Section 6.1: (1) If X = 0, then X has an element 1 0 / X ; (2) every part X i / Y has a complement X x i X /Y o i i / Y , i.e. X is a sum diagram. Now, we proceed with claim 3.4; X is not empty, so let 1 0 / X be an element and X i / Y a complement of i. The defining property of sum says that to define any / X (in particular the r we are looking for) is equivalent to specifying mapping Y mappings from the two parts whose sum is Y . We use the mappings
determining all of its J coordinates as being (essentially) all the given bonding maps f j . The other map p is simpler for it does not depend on the given bonding maps at all. Now if we are given any family of maps T ai / Ai for 1 i / I 3.7 Additional Exercises ai it can be considered as a single map T ai T By construction of p and f , ai I I / I Ai / I I // p f 75 Ai . Consider the diagram J Ac( j) will satisfy the equation p ai I = f ai I if and only if the given
of A0 ) such that X is the coproduct of A0 and / Y , f 1 : A1 / Y with codomain A1 ; i.e. for all Y , for any pair of arrows f 0 : A / Y there exists a unique f : X Y , making the diagram below commutative: 4.6 The Axiom of Choice: Distinguishing Constant/Random Sets A0 93 f0 i0 X f i1 A1 f1 It can be proved that the Boolean property follows from the axiom of choice. Now we only point out the randomness implied by the Boolean property: The maps f 0 , f 1 can be specified independently
picture, as follows. If A is an arbitrary abstract set mapped into the two element set by ϕ A · · · ·· ··• ·· · a0 ·· · · · · ·• · · · a1 ·· · ϕ 2 • • then A is divided in 2 parts. Conversely, any mutually exclusive and jointly exhaustive division of A into two parts arises from a mapping to the two-element set. (These are called either indicator functions or characteristic functions in probability theory, combinatorics, and other subjects.) The ϕ is the indicator or characteristic function