Let A={n ∈𝐙: 2 ≤|n|<4}, B={x ∈𝐐: 2<x ≤4}, C={x ∈𝐑: x^2-(2+√(2)) x+2 √(2)=0} and D={x ∈𝐐: x^2-(2+

step 1 In this question, we're going to list the members of each of these sets. So in A, I'm looking fo

Let A={n ∈𝐙: 2 ≤|n|<4}, B={x ∈𝐐: 2<x ≤4}, C={x ∈𝐑:...

Let $A=\{n \in \mathbf{Z}: 2 \leq|n|<4\}, B=\{x \in \mathbf{Q}: 2<x \leq 4\}$, $C=\left\{x \in \mathbf{R}: x^2-(2+\sqrt{2}) x+2 \sqrt{2}=0\right\}$ and $D=\left\{x \in \mathbf{Q}: x^2-(2+\sqrt{2}) x+2 \sqrt{2}=0\right\}$. (a) Describe the set $A$ by listing its elements. (b) Give an example of three elements that belong to $B$ but do not belong to $A$. (c) Describe the set $C$ by listing its elements. (d) Describe the set $D$ in another manner. (e) Determine the cardinality of each of the sets $A, C$ and $D$.

Let $A=\{n \in \mathbf{Z}: 2 \leq|n|<4\}, B=\{x \in \mathbf{Q}: 2<x \leq 4\}$,
$C=\left\{x \in \mathbf{R}: x^2-(2+\sqrt{2}) x+2 \sqrt{2}=0\right\}$ and $D=\left\{x \in \mathbf{Q}: x^2-(2+\sqrt{2}) x+2 \sqrt{2}=0\right\}$.
(a) Describe the set $A$ by listing its elements.
(b) Give an example of three elements that belong to $B$ but do not belong to $A$.
(c) Describe the set $C$ by listing its elements.
(d) Describe the set $D$ in another manner.
(e) Determine the cardinality of each of the sets $A, C$ and $D$.

Key Concepts

Set notation and intervals

Set notation is a mathematical language used to describe collections of objects. In this context, it represents sets defined by a rule or condition, such as a range of numbers or solutions to an equation. Interval notation is a specific way to denote a continuous set of numbers on the number line, where endpoints may or may not be included depending on the brackets used.

Absolute value

Absolute value is a measure of the distance of a number from zero, regardless of direction on the number line. It is crucial in defining sets that consider the magnitude of numbers rather than their sign, such as when specifying conditions like 2 <= |n| < 4.

Quadratic equations and their roots

Quadratic equations are polynomial equations of degree two. Solving a quadratic equation involves finding the roots, which are the values that satisfy the equation. The solutions might be found using methods such as factoring, completing the square, or the quadratic formula. Understanding the nature of these roots, including their rationality or irrationality, is essential in problems where sets are defined by the solutions of quadratic equations.

Domains of numbers (Integers, Rationals, and Reals)

The domains of numbers refer to different types of number sets: integers are whole numbers (including negatives, zero, and positives), rational numbers are numbers that can be expressed as fractions of integers, and real numbers include all numbers on the number line. Problems often require careful attention to these domains because the properties and permissible operations differ among them.

Cardinality

Cardinality is the concept of determining the size or number of elements in a set. It is important in set theory as it allows mathematicians to compare the sizes of finite and infinite sets, and to understand properties such as discreteness or continuity within a given set.

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