Infinite sets are mind-bending mathematical concepts that stretch our understanding of numbers. They contain endless elements, yet some can be counted while others can't. It's like trying to count grains of sand on a beach.
Dedekind-infinite sets take this idea further. They're so big that even if you remove some elements, you can still match them up with the original set. It's like having an infinite hotel where you can always make room for more guests.
Infinite Sets
Defining Infinite Sets
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Infinite set is a set that contains an infinite number of elements
Cannot be counted or enumerated completely, regardless of how much time is available
Cardinality , the size of an infinite set, is denoted by the symbol ℵ \aleph ℵ (aleph)
Two main types of infinite sets: countably infinite and uncountably infinite
Examples of Infinite Sets
Natural numbers N = { 1 , 2 , 3 , … } \mathbb{N} = \{1, 2, 3, \ldots\} N = { 1 , 2 , 3 , … } form a countably infinite set
Each natural number has a unique successor, creating an infinite sequence
Can be put into a one-to-one correspondence with the positive integers
Integers Z = { … , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , … } \mathbb{Z} = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\} Z = { … , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , … } also form a countably infinite set
Includes both positive and negative whole numbers, as well as zero
Can be mapped to the natural numbers by interleaving positive and negative integers (e.g., 0, 1, -1, 2, -2, ...)
Dedekind-Infinite Sets
Defining Dedekind-Infinite Sets
Dedekind-infinite set is an infinite set that contains a proper subset that can be put into a one-to-one correspondence with the original set
Proper subset is a subset that is not equal to the original set
One-to-one correspondence means that each element in one set can be paired with exactly one element in the other set, with no elements left unpaired
Properties of Dedekind-Infinite Sets
All Dedekind-infinite sets are infinite, but not all infinite sets are Dedekind-infinite
Removing a finite number of elements from a Dedekind-infinite set results in a set that is still Dedekind-infinite
Union of two Dedekind-infinite sets is also Dedekind-infinite
Cartesian product of two Dedekind-infinite sets is Dedekind-infinite
Types of Infinite Sets
Countably Infinite Sets
Rational numbers Q \mathbb{Q} Q form a countably infinite set
Can be expressed as fractions p q \frac{p}{q} q p , where p p p and q q q are integers and q ≠ 0 q \neq 0 q = 0
Can be put into a one-to-one correspondence with the natural numbers using a diagonalization argument (e.g., 1 1 , 1 2 , 2 1 , 1 3 , 2 2 , 3 1 , … \frac{1}{1}, \frac{1}{2}, \frac{2}{1}, \frac{1}{3}, \frac{2}{2}, \frac{3}{1}, \ldots 1 1 , 2 1 , 1 2 , 3 1 , 2 2 , 1 3 , … )
Algebraic numbers, roots of polynomials with integer coefficients, form a countably infinite set
Uncountably Infinite Sets
Real numbers R \mathbb{R} R form an uncountably infinite set
Includes rational and irrational numbers, which cannot be expressed as fractions
Cannot be put into a one-to-one correspondence with the natural numbers
Cardinality of the real numbers is denoted by c \mathfrak{c} c , the cardinality of the continuum
Cantor's theorem proves that the power set of any set has a strictly greater cardinality than the original set, implying the existence of uncountably infinite sets