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Problem 1

# Examine the following formal descriptions of sets so that you understand which members they contain. Write a short informal English description of each set. a. $$\\{1,3,5,7, \ldots\\}$$ b. $$\\{\ldots,-4,-2,0,2,4, \ldots\\}$$ c. $$\\{n \mid n=2 m$$ for some $$m$$ in $$\mathcal{N}\\}$$ d. $$\\{n \mid n=2 m$$ for some $$m$$ in $$\mathcal{N}$$, and $$n=3 k$$ for some $$k$$ in $$\mathcal{N}\\}$$ e. $$\\{w \mid w$$ is a string of os and 1 s and $$w$$ equals the reverse of $$w\\}$$ f. $$\\{n \mid n$$ is an integer and $$n=n+1\\}$$

Expert verified
a. Odd positive integers b. All even integers (positive, negative, and 0) c. Even positive integers d. Multiples of 6 e. Binary palindromes f. Empty set (∅)
See the step by step solution

## Step 1: a. Set A: {1, 3, 5, 7, ...}

This set contains all odd positive integers. They can be described as integers which are not divisible by 2.

## Step 2: b. Set B: {..., -4, -2, 0, 2, 4, ...}

This set contains all even integers, both positive and negative, as well as 0. They can be described as integers which are divisible by 2.

## Step 3: c. Set C: {n | n = 2m for some m in ℕ}

This set contains all the even positive integers. It contains the numbers obtained by multiplying any natural number m by 2.

## Step 4: d. Set D: {n | n = 2m for some m in ℕ, and n = 3k for some k in ℕ}

This set contains all positive integers that are both multiples of 2 and 3. These numbers can also be described as multiples of 6, since the least common multiple of 2 and 3 is 6.

## Step 5: e. Set E: {w | w is a string of 0s and 1s and w equals the reverse of w}

This set contains binary strings (strings made up of 0s and 1s) which read the same forwards and backwards. These strings are called palindromes.

## Step 6: f. Set F: {n | n is an integer and n = n + 1}

This set is actually empty (denoted by ∅), as it contains no elements. The given condition (n = n + 1) is not possible for any integer, as it would mean that 1 is equal to 0, which is a contradiction.

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