Math Problem Statement

Give one examples not drawn from the class notes of each of the following sets: (i) A Well Defined set (ii) A Property Defined set (iii) An infinite set (iv) A countably infinite set (v) Non Comparable Sets (vi) Non Disjoint Sets

Solution

Here are examples for each type of set:


(i) A Well-Defined Set

A set where its elements are clearly defined and there's no ambiguity in determining whether an object belongs to the set.
Example:
The set of all planets in the Solar System.
Notation:
S={Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune}S = \{ \text{Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune} \}


(ii) A Property-Defined Set

A set defined using a specific property or condition.
Example:
The set of all even integers greater than 10.
Notation:
T={xZx>10 and x is even}T = \{ x \in \mathbb{Z} \mid x > 10 \text{ and } x \text{ is even} \}


(iii) An Infinite Set

A set with an unbounded number of elements.
Example:
The set of all integers.
Notation:
I={,3,2,1,0,1,2,3,}I = \{ \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots \}


(iv) A Countably Infinite Set

An infinite set whose elements can be put into a one-to-one correspondence with the natural numbers.
Example:
The set of all rational numbers (fractions).
Notation:
Q={aba,bZ,b0}Q = \{ \frac{a}{b} \mid a, b \in \mathbb{Z}, b \neq 0 \}


(v) Non-Comparable Sets

Two sets that are neither subsets nor supersets of each other.
Example:
A={1,2,3}A = \{ 1, 2, 3 \} and B={a,b,c}B = \{ a, b, c \}.
Here, AA and BB are non-comparable because they share no elements.


(vi) Non-Disjoint Sets

Two sets that have at least one element in common.
Example:
X={1,2,3}X = \{ 1, 2, 3 \} and Y={3,4,5}Y = \{ 3, 4, 5 \}.
Here, XY={3}X \cap Y = \{ 3 \}, so they are non-disjoint.


Do you want details or have any questions?

5 Related Questions:

  1. How do you determine if a set is countably infinite versus uncountably infinite?
  2. What are some real-world applications of property-defined sets?
  3. Can a set be both infinite and well-defined? Provide examples.
  4. What is the difference between disjoint and comparable sets?
  5. How can you visually represent the relationship between non-comparable sets?

Tip:

When working with infinite sets, remember to distinguish between countable and uncountable infinities—this is crucial in higher mathematics!

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Math Problem Analysis

Mathematical Concepts

Set Theory
Well-Defined Sets
Property-Defined Sets
Infinite Sets
Countably Infinite Sets
Non-Comparable Sets
Non-Disjoint Sets

Formulas

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Theorems

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Suitable Grade Level

Grades 9-12