Math 311-001 201110 Assignment # 1 (due: Jan 27th) Summary: Math 311-001 201110 Assignment # 1 (due: Jan 27th) 1. Assume the existence of N with its addition and multiplication. (a) Complete the details for the construction of Z (i.e. show that there exists a commutative ring whose elements are exactly the natural numbers, their additive inverses, and zero; show that addition and multiplication extend to Z and they satisfy axioms I-VI (with the exception of multiplicative inverses). (b) Complete the details for the construction of Q (i.e. show that there exists a field satisfying I-VI, that contains Z and such that every element is an integer multiple of the multiplicative inverse of an integer). (c) Complete the details for the construction of R (i.e. show that there exists a field satisfying I-VII) and prove that such a field is unique (up to isomorphism of fields, i.e. if there is another field R satisfying I-VII, then there is a bijection R R that preserves sums and products). 2. Consider the space R × R, and the function d((x, y), (z, w)) = |x - z| + |y| + |w| if x = z Collections: Mathematics