Equivalence Classes

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Equivalence Classes

From section 7.3 of the book and this equivalence class discussion.

Examples

Consider the relation ~ on the natural numbers, defined by a ~ b if and only if ⌊ log₂a ⌋ = ⌊ log₂b ⌋. Convince yourself that ~ is an equivalence relation. Describe the equivalence classes of ~.

This is an equivalence relation because any number has the same logarithm as itself (reflexivity), if ⌊ log₂a ⌋ = ⌊ log₂b ⌋ then ⌊ log₂b ⌋ = ⌊ log₂a ⌋ (symmetry), and if ⌊ log₂a ⌋ = ⌊ log₂b ⌋ and ⌊ log₂b ⌋ = ⌊ log₂c ⌋ then ⌊ log₂a ⌋ = ⌊ log₂c ⌋ (transitivity).

The equivalence classes (i.e., sets of numbers that are equivalent in this relation) are intervals in the natural numbers:

Notice that each interval is twice as long as the one before. If the equivalence relation were based on base-3 logarithms, the intervals would triple in length, etc.

Imagine a set of files divided up into folders on a computer. Are there equivalence classes in that organization? If so, of what equivalence relation?

If you think of an “in same folder” equivalence relation, then the files in each folder are equivalence classes. 

Properties

Can an equivalence class be the empty set? No. For an equivalence relation on set A, every x ∈ A is in [x], so no equivalence class is ever empty.

For an equivalence relation on some set A, can all of A be a single equivalence class? Yes. For example, let A = {1} and R = { (1,1) }.

Can an equivalence class have only 1 member? Yes. The above example illustrates this.

Equivalence Classes and Partitions

Let A be some non-empty set.

Why does every partition of A define an equivalence relation on A?

A partition of A is a way of dividing A into non-empty, disjoint, subsets such that every member of A is in one of the subsets. For example

Think about an “in same subset” relation on A defined by a partition.

• It’s reflexive: every element of A is in some subset and it’s the same subset as itself.
• Symmetric: If a and b are in the same subset, then so are b and a.
• Transitive: If a and b are in same subset and so are b and c, then a and c are.

Would this still be true if we relaxed some of the properties of partitions? For example…

If the subsets in a partition didn’t have to be disjoint. Then the “in same subset” relation wouldn’t necessarily be transitive. You could have a and b in subset A₁, b and c in A₂, so a and b are in the same subset, and b and c are, but a and c are not in same subset.

If some of the subsets in a partition could be empty. Then you could still define an “in same subset” equivalence relation from the non-empty subsets.

Next

Start talking about infinite sets, and through them the notion of infinity. We need two things in order to do this:

1. A way to talk about the sizes of sets that are too big to just have their members counted, and
2. An understanding of finite sets (since infinite sets are just ones that aren’t finite).

Please read section 9.1 in the textbook.

Please also contribute to this discussion of equivalent sets.

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