Relevant definitions:
THE EMPTY SET: “There is a special set that has no elements. This set is called the empty set, or null set, and is denoted by ∅. The empty set can also be denoted by { } (that is, we represent the empty set with a pair of braces that encloses all the elements in this set). Often, a set of elements with certain properties turns out to be the null set. For instance, the set of all positive integers that are greater than their squares is the null set.”
“A set with one element is called a singleton set.”
“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.”
Theorem: “For every set S, (i) ∅⊆S and (ii)S ⊆S.”
“Two sets are equal if and only if they have the same elements. Therefore, if A and B are sets, then A and B are equal if and only if∀x(x ∈A↔x ∈B). We write A=B if A and B are equal sets.”
The definitions and theorem above are from the textbook Discrete Mathematics and its Applications by Rosen.
Table of Contents
- a) ∅∈{∅}
- b) ∅∈{∅,{∅}}
- c) {∅} ∈ {∅}
- d) {∅} ∈ {{∅}}
- e) {∅} ⊂ {∅,{∅}}
- f) {{∅}} ⊂ {∅, {∅}}
- g) {{∅}}⊂{{∅},{∅}}
a) ∅∈{∅}
We can see from that the singleton set {∅} is the set with one element: ∅.
Therefore, this statement is true.
b) ∅∈{∅,{∅}}
This statement is true.
c) {∅} ∈ {∅}
The set in the right {∅} has only one element: ∅, and ∅≠{∅}.
Therefore, this statement is false.
d) {∅} ∈ {{∅}}
This statement is true.
e) {∅} ⊂ {∅,{∅}}
Using the definition of subset, we can see that every element in the first set (only one element ∅ in this case), is also an element in the second set. Also, the first set is not equal to the second one as the second one has one element, {∅}, that is not an element of the first one.
Therefore, this statement is true.
f) {{∅}} ⊂ {∅, {∅}}
Using the definition of subset, we can see that every element in the first set (only one element {∅} in this case), is also an element in the second set.
Also, there is an element in the second set ∅, that is not an element of the first set. So, these two sets are not equal.
A set A is a proper subset of a set B if all elements of A are also elements of B and A≠B.
Therefore, this statement is true.
g) {{∅}}⊂{{∅},{∅}}
Using the definition of subset, the only element, {∅}, of the first set is also an element of the second set. However, the first set and the second one are equals. A set A is a proper subset of a set B if all elements of A are also elements of B and A≠B.
Therefore, this statement is false.
A word of caution. To make sure you don’t make mistakes in this type of exercise, make sure to notice that there is a difference between finding when an element belongs to a set (∈) and when a set is a subset of another set (⊆,⊂)
Related exercises:
- Exercises 1 to 5 from the textbook
- 6. Suppose that A = {2,4,6}, B = {2,6}, C = {4,6}, and D = {4, 6, 8}. Determine which of these sets are subsets of which other of these sets
- 8. For each of the sets in Exercise 7, determine whether {2} is an element of that set
- 7. For each of the following sets, determine whether 2 is an element of that set
- 9. Determine whether each of these statements is true or false
