Write a specification by rules and one by predicates for...

Write a specification by rules and one by predicates for each of the following sets. Remember that there is no order assumed in the list, so you cannot use words like 'the first' or 'the latter'. Recall also that a recursive rule may contain more than one if-then statement. (a) $\{5,10,15,20, \ldots\}$ (b) $\{7,17,27,37, \ldots\}$ (c) $\{300,301,302, \ldots, 399,400\}$ (d) $\{3,4,7,8,11,12,15,16,19,20, \ldots\}$ (e) $\{0,2,-2,4,-4,6,-6, \ldots\}$ (f) $\{1,1 / 2,1 / 4,1 / 8,1 / 16, \ldots\}$

Write a specification by rules and one by predicates for each of the following sets. Remember that there is no order assumed in the list, so you cannot use words like 'the first' or 'the latter'. Recall also that a recursive rule may contain more than one if-then statement.
(a) $\{5,10,15,20, \ldots\}$
(b) $\{7,17,27,37, \ldots\}$
(c) $\{300,301,302, \ldots, 399,400\}$
(d) $\{3,4,7,8,11,12,15,16,19,20, \ldots\}$
(e) $\{0,2,-2,4,-4,6,-6, \ldots\}$
(f) $\{1,1 / 2,1 / 4,1 / 8,1 / 16, \ldots\}$

Key Concepts

Recursive Specification

A recursive specification defines a set by establishing one or more base elements and providing rules that allow new elements to be generated from previously defined ones. This method relies on the principle of mathematical induction, where the base case starts the set and the recursive step expands it, ensuring that every element in the set can be reached through finite applications of the rule.

Predicate Specification

A predicate specification characterizes a set by a property or condition that every element in the set must satisfy. Instead of describing the set constructively, this approach uses a logical formula or condition to define membership, so an element belongs to the set if and only if it fulfills the given predicate.

Arithmetic Sequences

Arithmetic sequences consist of numbers with a constant difference between consecutive terms. These sequences can be specified either recursively (by adding a fixed number to a starting term) or by an explicit formula representing each term’s position. This concept is essential when defining sets that progress uniformly.

Geometric Sequences

Geometric sequences consist of numbers where each term is obtained by multiplying the previous term by a constant ratio. They can be defined recursively with a base element and a multiplicative rule or by using an explicit formula, and are particularly useful for describing sets that shrink or grow exponentially.

Bounded Intervals

Bounded intervals refer to collections of consecutive integers confined between a specified lower and upper bound. In such intervals, every integer within the boundary is included, leading to a finite set. This concept is often used to describe sets that are limited to a specific range rather than extending indefinitely.

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