By William Parr Greswell

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6 onto We may also consider the relation {(x, x) | x ∈ A}, denoted IA , where A is a proper subset of the given space S. Then IA : A → S is one-to-one but not onto S (it is onto A only). IA is called the identity map on A. §5. Mappings 25 For, if f is one-to-one, then no distinct elements of its domain can have one and the same function value y. , that f −1 is a function. The function value f (x) is also sometimes denoted by f x, xf , or fx . In the latter case (called “index notation”), the domain of f is also referred to as an index set, and the range of f is denoted by {fx }.

A21 a22 a23 . . . a2n . . a31 a32 a33 . . . a3n . . an1 an2 an3 . . . ann . . Therefore the method used above is called the diagonal process (due to Cantor). Now, by Corollary 2, all rationals can be put in a sequence. But, as shown above, no such sequence can cover all of [0, 1). , cannot be written as ratios of integers, n/m. Moreover, such numbers, called irrational, must form an uncountable set, for otherwise its union with the countable set of all rationals in [0, 1) would be countable (by Note 2), whereas actually this union is the uncountable set [0, 1).

What are its inverse, domain, and range? What if A = ∅ or B = ∅? How many elements (ordered pairs) does A × B contain if A has m elements and B has n elements (both finite)? 5 15. Prove the following identities, and illustrate by diagrams. ) (i) (A ∪ B) × C = (A × C) ∪ (B × C); ∗ (ii) (A ∩ B) × (C ∩ D) = (A × C) ∩ (B × D); (iii) (X × Y ) − (X ′ × Y ′ ) = [(X ∩ X ′ ) × (Y − Y ′ )] ∪ [(X − X ′ ) × Y ]. 16. Prove the following: (i) (A × B) ∩ (C × D) = ∅ iff A ∩ C = ∅ or B ∩ D = ∅; (ii) A × B = C × D iff each product has ∅ as one of the factors or A = C and B = D; 5 In this and the following problems, we shall be satisfied with the intuitive notion of a finite set and the number of its elements.

### Annals of Parisian Typography, Containing an Account of the Earliest Typographical Establishments of Paris; and Notices and Illustrations, of the Most Remarkable Productions of the Parisian Gothic Press: Compiled Principally to Shew Its General Character by William Parr Greswell

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