Question

For each of the following pairs of sets, prove that they are equinumerous. Remember that we have two ways to do this: we can find a bijection explicitly, or we can prove that there is an injection in each direction and then use the Schr¨oder-Bernstein theorem.

4. N and Qd for d > 1 5. R and R x R {a + bi |2 =-1, a,bE R} is the complex numbers) 6. R and C (where C

Answer 1

Definition (1) A set A is said to be countable if exist bijection from a subset of N to A or, (2) exist a 1 - 1 function from A to N (3) vert A vert denotes the candinality of the set A. (4) To show: vert N vert = -vert Q^d vert: d greaterthanorequalto 1 1st step: vert N vert = vert Q^+ vert let, A_1 = {1/1, 1/2, 1/3, ----} A_2 = {2/1, 2/2, 2/3, ----} vertical dots A_i = {i/1, i/2, i/3, -----} Then, Union_i elementof N A_i = Q^+ Let, f_i: N rightarrow A_i by n map i/n Then, f_i is a bijection rightwardsdoublearrow vert N vert = vert A_i vert rightwardsdoublearrow A_u is countable and A = Union_i elementof N A_i: countable union of countable sets rightwardsdoublearrow A is countable rightwardsdoublearrow vert N vert = vert A vert = vert Q^+ vert Similarly, Q^- is also countable. Now, Q = Q^+ Union Q^- Union{0} = countable