7. For each of the following sets, determine whether 2 is an element of that set

Relevant definitions:

“A set is an unordered collection of objects, called elements or members of the set. A set is said to contain its elements. We write a ∈ A to denote that a is an element of the set A. The notation a∉A denotes that a is not an element of the set A.”

“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.”

“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.”

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

a) {x ∈ R|x is an integer greater than 1}
b) {x ∈ R|x is the square of an integer}
c) {2,{2}}
d) {{2},{{2}}}
e) {{2},{2,{2}}}
f ) {{{2}}}

a) {x ∈ R|x is an integer greater than 1}

This set is specified using the builder notation. So, we need to check if the element 2 have the properties specified for this set.

The property specified for all the elements of this set is that they must be greater than 1. 2 is greater than 1.

Therefore, the element 2 is an element of this set.

b) {x ∈ R|x is the square of an integer}

Following a similar approach as in the previous exercise, we need to check whether the element 2 has the property specified in the set.

Is 2 the square of an integer? To answer this, we can calculate the square root of 2. This will give us a number x, such that x²=2.

√2=1.41

1.41 is not an integer number.

Therefore, the element 2 is not an element of this set.

c) {2,{2}}

This set is specified by the roster method (see relevant definitions section above).

So, in this case, we just need to compare each element of this set with 2.

We can see that 2 is an element of this set.

d) {{2},{{2}}}

In this case, the set is also specified using the roster method.

The set has only two elements: {2} and {{2}}.

{2} is the set with 2 as the only one element, {2}≠2.

{{2}} is the set that has only one element. The element is a set with 2 as the only element, {{2}}≠2.

Therefore, 2 is not an element of this set.

e) {{2},{2,{2}}}

Following a similar reasoning as in the previous exercise, we can conclude that 2 is not an element of this set.

f ) {{{2}}}

This set has only one element: {{2}}.

The element {{2}} is not equal to 2.