For each of the following sets, determine whether 2 is an...

For each of the following sets, determine whether 2 is an element of that set. a) $\{x \in \mathbf{R} \mid x$ is an integer greater than 1$\}$ b) $\{x \in \mathbf{R} \mid x$ is the square of an integer $\}$ c) $\{2,\{2\}\}$ d) $\{\{2\},\{\{2\}\}\}$ e) $\{\{2\},\{2,\{2\}\}\}$ f) $\{\{\{2\}\}\}$

For each of the following sets, determine whether 2 is an element of that set.
a) $\{x \in \mathbf{R} \mid x$ is an integer greater than 1$\}$
b) $\{x \in \mathbf{R} \mid x$ is the square of an integer $\}$
c) $\{2,\{2\}\}$
d) $\{\{2\},\{\{2\}\}\}$
e) $\{\{2\},\{2,\{2\}\}\}$
f) $\{\{\{2\}\}\}$

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Key Concepts

Set Membership

Set membership is a fundamental concept in set theory that describes the relationship between an element and a set, denoted by the symbol ?. It addresses whether a given object is one of the items contained directly within the set, making it essential to determine if a particular object qualifies as a member of a set.

Set-builder Notation

Set-builder notation provides a concise way to define a set by specifying a property or condition that its elements must satisfy, rather than enumerating its elements explicitly. This notation is useful for describing infinite or large sets and clarifies the criteria for membership.

Nested Sets

Nested sets are sets that have other sets as elements. This concept requires a careful analysis of the levels of containment, distinguishing between an element that is a set containing certain objects and the objects themselves. Understanding nested sets is critical when elements appear at different levels of the set hierarchy.

Singleton Sets

A singleton set is a set that contains exactly one element. It is important to recognize that the singleton set {a} is distinct from the element a itself, which plays a vital role in problems where distinguishing between an element and a set that includes that element is necessary.

Transcript

00:01 For each of the following sets, we are asked to determine whether or not 2 is an element of that set.

00:10 First, we have the set x such that x is an integer greater than 1.

00:15 We can list a few elements of this set.

00:18 It's strictly greater than 1, so the first element is going to be 2, 3, 4, 5, and all integers following after that.

00:28 Call this set a.

00:29 Well, we can see here that 2 is definitely a member of set a.

00:35 Next, we have the set x such that x is the square of an integer.

00:44 So, we have the set of integers ranging from negative infinity to infinity, and we have this set as the squares of all of those integers, right? since the negative squares are going to equal the positive squares, let's just start at 0.

01:03 0 squared is 0, 1 squared is 1, 2 squared is 4, 3 squared is 9, and so on.

01:12 We can see here that there is no such x such that 2 is the square of an integer, right? there is no integer such that x squared is equal to 2.

01:26 So, if this is our set b, 2 is not an element of the set b.

01:37 Next, we have the set c here, which contains 2 and the set containing 2.

01:47 So, let's look at this set here.

01:49 We have 2 members, and surely one of those members is 2, so 2 is an element of c.

01:59 Next, we have our set d here, which contains.....