9. Determine whether each of these statements is true or false

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.”

The definitions and theorem above are from the textbook Discrete Mathematics and its Applications by Rosen.

Table of Contents

• a) 0∈∅
• b) ∅∈{0}
• c) {0}⊂∅
• d) ∅⊂{0}
• e) {0}∈{0}
• f) {0}⊂{0}
• g) {∅}⊆{∅}

a) 0∈∅

The null set has no elements.

Therefore, this statement is false.

b) ∅∈{0}

The null set is not an element of the set {0}, as the last one has only one element: 0.

Therefore, this statement is false.

c) {0}⊂∅

The null set has no elements. So, a set with 1 element cannot a subset of the null set.

Therefore, this statement is false.

d) ∅⊂{0}

Using the theorem above, we can answer that this statement is true.

e) {0}∈{0}

The set element {0} does not belong to the set {0}. In other words, the set {0} has only one element 0, and 0≠{0}.

Therefore, this statement is false.

f) {0}⊂{0}

The symbol ⊂ means proper subset. A set A is a proper subset of the set B if and only if A is a subset of B and A≠B.

In this case, A=B.

Therefore, this statement is false.

g) {∅}⊆{∅}

All the elements of the first set, which is the null set, are also elements of the second set.

Therefore, this statement is true.

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
• 10. Determine whether these statements are true or false