If A={x ∈R:|x|<2} and B={x ∈R:|x-2| ≥3} ; then : (a) A ∩B=(-2,-1) (b) B-A=R-(-2,5) (c) A ∪B=R-(2

step 1 In this question, we are given two sets. The first set is A, which says that X belongs to real n

If A={x ∈R:|x|<2} and B={x ∈R:|x-2| ≥3} ; then : (a) A...

If $\mathrm{A}=\{x \in \mathrm{R}:|x|<2\}$ and
$\mathrm{B}=\{x \in \mathrm{R}:|x-2| \geq 3\} ;$ then :
(a) $\mathrm{A} \cap \mathrm{B}=(-2,-1)$
(b) $\mathrm{B}-\mathrm{A}=\mathrm{R}-(-2,5)$
(c) $\mathrm{A} \cup \mathrm{B}=\mathrm{R}-(2,5)$
(d) $A-B=[-1,2)$

Key Concepts

Absolute Value Inequalities

Absolute value inequalities involve expressions with the absolute value function, which measures the distance from zero on the real line. Solving these inequalities requires understanding the definition of absolute value, which leads to splitting the inequality into two cases. This concept is fundamental in determining the set of all real numbers that satisfy a given distance condition from a point, as seen with conditions like |x| < c or |x - a| ? b.

Set Intersection

Set intersection is a key operation in set theory that finds the common elements between two sets. It involves identifying all points that simultaneously satisfy the conditions defining both sets. This concept is crucial when combining multiple conditions on real numbers, such as finding the overlap between two intervals or regions defined by respective inequalities.

Set Union

Set union is a basic operation in set theory that combines the elements of two or more sets, including any element that appears in at least one of the sets. In the context of real numbers, union allows for the aggregation of different intervals or regions, especially when dealing with solutions of inequalities that need to be considered together.

Set Difference

Set difference refers to the operation of removing elements from one set that are also contained in another. It is useful when one needs to isolate a subset of elements that satisfy one condition, but not another. This concept helps in excluding overlapping regions from a given interval, which is important in characterizing parts of the real number line.

Interval Notation

Interval notation is a concise way of representing sets of real numbers defined by inequalities. It provides a clear and standardized method to depict intervals, using parentheses and brackets to indicate whether endpoints are excluded or included. This notation is essential for communicating the exact ranges of numbers that satisfy certain conditions, making it a fundamental tool in the study of real analysis and set theory.

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