# Set Theory and Counting

1. List the ordered pairs in the equivalence relations produced by these partitions of {0,1,2,3,4,5}

a) {0}, {1,2}, {3,4,5}

b) {0,1}, {2,3}, {4,5}

c) {0,1,2}, {3,4,5}

d) {0}, {1}, {2}, {3}, {4}, {5}

2. Which of these collections of subsets are partitions of the set of integers?

a) the set of even integers and the set of odd integers

b) the set of positive integers and the set of negative integers

4. Which of these collections of subsets are partitions of {1,2,3,4,5,6}?

a) {1,2}, {2,3,4}, {4,5,6}

b) {1}, {2,3,6}, {4}, {5}

c) {2,4,6}, {1,3,5}

d) {1,4,5}, {2,6}.

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#### Solution Preview

Solution to Q1.

(a)

- As {0} is an equivalence class, 0 is related to 0. Hence, (0,0) is the only ordered pair corresponding to the equivalence class {0}.

- As {1,2} is an equivalence class, there are four ordered pairs (1,1), (1,2), (2,1), (2,2).

- As {3,4,5} is an equivalence class, there are nine ordered pairs (3,3), (3,4), (4,3), (4,4), (3,5), (5,3), (4,5),(5,4), (5, 5). So, the ordered pairs in the equivalence relation are {(0,0), (1,1), (1,2), (2,1), (2,2), (3,3), (3,4), (4,3), (4,4), (3,5), (5,3), (4,5),(5,4), (5, 5)}

(b)

- As {0,1} is an equivalence class, there are four ordered pairs (0, 0), (0,1), (1,0), (1,1).

- ...

#### Solution Summary

The solution discusses the set theory and counting.

A Set Theory in Real Analysis

Formal Math Proofs

Prove that each of the following sets is countable:

a) The set of all numbers with two distinct decimal expansions (like 0.500... and 0.4999...);

b) The set of all rational points in the plane (i.e., points with rational coordinates);

c) The set of all rational intervals (i.e., intervals with rational end points);

d) The set of all polynomials with rational coefficients.

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