The set T={2 k+1: k ∈𝐙} can be described as T={…,-3,-1,1,3, …}. Describe the following sets in a

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The set T={2 k+1: k ∈𝐙} can be described as T={…,-3,-1,1,3,...

The set $T=\{2 k+1: k \in \mathbf{Z}\}$ can be described as $T=\{\ldots,-3,-1,1,3, \ldots\}$. Describe the following sets in a similar manner.
(a) $A=\{4 k+3: k \in \mathbf{Z}\}$
(b) $B=\{5 k-1: k \in \mathbf{Z}\}$.

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Key Concepts

Modular Arithmetic

Modular arithmetic involves numbers and their remainders upon division by a fixed number (the modulus). For a set defined by a formula like ak + b, the elements are all those numbers that leave a remainder of b when divided by a (or a related modulus computation). This concept helps in re-describing the set in terms of congruence classes; for example, the set {4k + 3: k ? ?} consists of all integers that are congruent to 3 modulo 4.

Set-builder Notation

Set-builder notation is a concise way of specifying a set by stating the properties its members must satisfy. When we write something like {ak + b: k ? ?}, we mean all numbers that can be produced by this linear formula. Understanding set-builder notation is key to translating and describing sets in different forms, such as listing out several elements of the sequence or describing them by their modular congruence.

Arithmetic Progression

An arithmetic progression is a sequence of numbers where the difference between consecutive terms is constant. In the context of sets defined by a linear expression such as {ak + b: k ? ?}, the sequence forms an arithmetic progression with a common difference of a. This concept is crucial because it allows us to recognize that such sets are infinite and have regularly spaced elements.

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