Which of the following sets are equal? A={n ∈𝐙:|n|<2} D={n ∈𝐙: n^2 ≤ 1} B={n ∈𝐙: n^

step 1 Hello students in this question we are given several sets and we need to find which of them are

Which of the following sets are equal? A={n ∈𝐙:|n|<2} ...

Which of the following sets are equal?
$$
\begin{array}{ll}
A=\{n \in \mathbf{Z}:|n|<2\} & D=\left\{n \in \mathbf{Z}: n^2 \leq 1\right\} \\
B=\left\{n \in \mathbf{Z}: n^3=n\right\} & E=\{-1,0,1\} . \\
C=\left\{n \in \mathbf{Z}: n^2 \leq n\right\} &
\end{array}
$$

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Key Concepts

Solving Polynomial Equations

Many sets are defined by equations like n^3 = n, which require solving a polynomial to determine the possible values of n. Techniques such as factoring and setting factors equal to zero are central to this process.

Absolute Value and Inequalities

Absolute value inequalities, such as those involving |n|, restrict elements based on their distance from zero. This concept helps define specific intervals within the set of integers or real numbers and is essential for understanding sets bounded by a fixed magnitude.

Quadratic Inequalities

Inequalities involving quadratic expressions, such as n^2 ? n or n^2 ? 1, are solved by determining the range of values that satisfy the condition. Analyzing these inequalities involves understanding the shape of quadratic functions and their roots.

Set Equality

Two sets are considered equal if every element of one set is also an element of the other, and vice versa, regardless of the order in which the elements are listed. This concept is fundamental in set theory and underlies the process of comparing sets defined in different ways.

Set Builder Notation

A notation used to define a set by specifying the properties that its members must satisfy. This method allows for the concise representation of sets and is a standard language for describing classes of objects in mathematics.

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