1. State both forms of the Archimedean property of R. This is Theorem 1.4.2: (i) Given any real number x, there is a natural number n such that n > x. (ii) Given any real number y > 0, there is a natural number n such that 1=n < y. 2. What do we mean when
1. What does it mean to say that two sets A and B have the same cardinality? What does it mean to say that a set S is countable? The .rst is De.nition 1.4.7 in the text: Two sets A and B have the same cardinality if there exists f : A ! B that is one-to-o
1. Let A be a subset of R and s a real number. (a) De.ne what it means for s to be an upper bound for A. De.nition 1.3.1 in the text says that s is an upper bound for A if a s for every element a in A. (b) De.ne what it means for s to be the least upper b
In the .rst three questions, f is a function from R to R. 1. For A a subset of R, (a) De.ne the range, f (A), of f over A. f (A) = ff (x) : x 2 Ag (b) De.ne the preimage, f f
1 1
(A), of A.
(A) = fx : f (x) 2 Ag
1
2. Show that if A and B are subsets of R,
1. What is a rational number ? Abbott says, "A rational number is any number that can be expressed in the form p=q where p and q are integers." You can call p=q a ratio or a quotient or a fraction if you want. You can add that q 6= 0. But you can' require