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Chapter 6: Problem 77

Explain the meaning of a universal set, and give two different universal sets that could be used in a discussion about sets of positive integers.

Short Answer

Expert verified
A universal set is the set containing all objects or elements under consideration in a particular context or problem, and every other set discussed is assumed to be a subset of this universal set. Two examples of universal sets related to sets of positive integers are: 1. The set of all positive integers, denoted as \(\mathbb{N}\), which includes all natural numbers greater than 0: \[\mathbb{N} = \{1, 2, 3, 4, 5, ...\}\] 2. The set of all even positive integers, defined as \[E = \{2, 4, 6, 8, ...\}\]

Step by step solution

01

Define universal set

A universal set is the set containing all objects or elements under consideration in a particular context or problem, and every other set discussed is assumed to be a subset of this universal set. In other words, it is the set that contains all the elements relevant to the problem being discussed.
02

Provide example 1 of a universal set related to positive integers

The set of all positive integers can be an example of a universal set in certain contexts. This set is denoted as \(\mathbb{N}\), and contains all natural numbers greater than 0: \[ \mathbb{N} = \{1, 2, 3, 4, 5, ...\}\] In discussions about sets of positive integers, this universal set would contain all the numbers that we are considering.
03

Provide example 2 of a universal set related to positive integers

Another example of a universal set related to positive integers could be the set of all even positive integers. This set is defined as \[E = \{2, 4, 6, 8, ...\}\] In discussions related to sets of even positive integers, this universal set would include all the relevant numbers being considered.

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