Sets is one of the basic structures in Discrete Mathematics. Here, you will find the solution to 5 set exercises from one of the more used textbooks in this topic.
Table of Contents
- Exercise 1. List the members of the following sets
- Exercise 2. Use set builder notation to give a description of each of these sets
- Exercise 3. For each of these pairs of sets, determine whether the first is a subset of the second, the second is a subset of the first, or neither is a subset of the other
- Exercise 4. For each of these pairs of sets, determine whether the first is a subset of the second, the second is a subset of the first, or neither is a subset of the other
- Exercise 5. Determine whether each of these pairs of sets are equal
Exercise 1. List the members of the following sets
In this case, we have to use the roaster method.
” There are several ways to describe a set. One way is to list all the members of a set, when this is possible. We use a notation where all members of the set are listed between braces. For example, the notation {a, b, c, d} represents the set with the four elements a, b, c, and d. This way of describing a set is known as the roster method. “
Discrete Mathematics and its Applications by Rosen
a) {x|x is a real number such that x2 =1}
{-1,1}. Notice that -1 and 1 are the only real numbers that when are squared they equals 1.
b) {x | x is a positive integer less than 12}
{1,2,3,…,11}. In this case, we just have to list all positive integers that are less than 12, we can use … when the pattern is obvious, so we don’t have to write every single element of the set.
c) {x|x is the square of an integer and x<100}
{1,4,9,16,25,36,49,64,81}. Notice the elements of the previous set are the only ones that are less than 100 and the square root is an integer number.
d) {x|x is an integer such that x2 =2}
{} (the empty set, also known as the null set, can also be denoted by ∅). In this case, the result is the empty set. One can be confused and state that the square root 2 is a member of the set, but that is a real number, not an integer; therefore, it does not belong to that set.
Also notice here, that the empty set {} or ∅, is not the same the set {∅}. The first set has no elements. The last set is the set with one element, in this case the element of that set is the empty set.
” The empty set can be thought of as an empty folder and the set consisting of just the empty set can be thought of as a folder with exactly one folder inside, namely, the empty folder. “
Discrete Mathematics and its Applications by Rosen
Exercise 2. Use set builder notation to give a description of each of these sets
In this exercise, we have to do the opposite than the previous one.
“Another way to describe a set is to use set builder notation. We characterize all those elements in the set by stating the property or properties they must have to be members. “
Discrete Mathematics and its Applications by Rosen
a) {0,3,6,9,12}
{x | x is a non-negative integer less than 13 and divisible by 3}. In this case, we need to include 0 (non-negative integer), the property all of them have is they are divisible by 3 and they are less or equal than 12 (or less than 13).
b) {x | integers greater than -4 and less than 4}. Notice all the elements of the set are integers from -3 to 3.
{−3,−2,−1,0,1,2,3}
c) {m, n, o, p}
{x | English alphabet letters from m to p}.
Usually, in this type of exercise there is no unique answer. It means the answers bellow are not the only correct answers.
Exercise 3. For each of these pairs of sets, determine whether the first is a subset of the second, the second is a subset of the first, or neither is a subset of the other
In mathematics, it is very important to know the definitions. It is the only way of doing things right.
Let’s start by reading the definition of subset.
” The set A is a subset of B if and only if every element of A is also an element of B. We use the notation A ⊆ B to indicate that A is a subset of the set B. “
Discrete Mathematics and its Applications by Rosen
Using the definition above, we can easily answer the questions.
a) the set of airline flights from New York to New Delhi, the set of nonstop airline flights from New York to New Delhi
a) No. Because there are elements of the first subset (A) that are not in the second subset (B), they are the airline flights from New York to Delhi with stops.
b) the set of people who speak English, the set of people who speak Chinese
b) No. Because there are elements of the first subset (A) that are not in the second subset (B), they are the people who speaks English and don’t speak Chinese (myself).
c) the set of flying squirrels, the set of living creatures that can fly
Yes. Because all the elements from the first set (flying squirrels) also belongs to the second set (living creatures that can fly).
Exercise 4. For each of these pairs of sets, determine whether the first is a subset of the second, the second is a subset of the first, or neither is a subset of the other
To answer this question, we have to use the definition of subset.
” The set A is a subset of B if and only if every element of A is also an element of B. We use the notation A ⊆ B to indicate that A is a subset of the set B. “
Discrete Mathematics and its Applications by Rosen
a) the set of people who speak English, the set of people who speak English with an Australian accent
The second is subset of the first one. Because each people who speak English with Australian accent (elements of the second set), also speak English (elements of the first set).
b) the set of fruits, the set of citrus fruits
The second is subset of the first one. Each citrus fruit (element of the second set) is also a fruit (element of the first set).
c) the set of students studying discrete mathematics, the set of students studying data structures
None of the two are subset of the other one. A student can be studying Discrete Mathematics and not studying data structures. Also, a student can be studying data structures and not studying Discrete Mathematics.
Exercise 5. Determine whether each of these pairs of sets are equal
Now we have to solve another type of exercise. As I always recommend, let’s first look at the definition.
“To show that two sets A and B are equal, show that A ⊆ B andB ⊆A. “
“ Showing that A is a Subset of B” To show that A⊆B, show that if x belongs to A then x also belongs to B.
Showing that A is Not a Subset of B: To show that A/⊆B, find a single x∈A such that xÏB.”
Discrete Mathematics and its Applications by Rosen
a) {1,3,3,3,5,5,5,5,5},{5,3,1}
Yes. Because every element of the first set belongs to the second set and, every element of the second set belongs to the first set.
b) {{1}},{1,{1}}
No. The element 1 (don’t get confused with the element {1}) belongs to the second set and it does not belong to the first set.
c) ∅,{∅}
No. The first set is the empty set (no elements). The second set is a set with one element. Therefore, they are not the same.
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