Question

Prove or disprove: If \(R\) is an equivalence relation on an infinite set \(A,\) then \(R\) has infinitely many equivalence classes.

Short answer

The statement is false. We can find a counterexample in the set of integers with the equivalence relation defined by whether the difference is an even integer. Even though the set (the integer set) is infinite, the equivalence classes (the set of even integers and the set of odd integers) are not infinite.

Step-by-step solution

Identify the statement

The statement to be proven is 'If \(R\) is an equivalence relation on an infinite set \(A,\) then \(R\) has infinitely many equivalence classes.' We need to either prove this statement true or prove that it's false by providing a counterexample.

Counterexample for disproving

To disprove the statement, it's enough to find one counterexample. Consider an equivalence relation \(R\) on set \(A\) as defined by \(a R b\) if \(a - b\) is an even integer. This is an equivalence relation on any infinite set \(A\) of integers according to the properties reflexivity (\(aRa\)), symmetry (\(aRb \Rightarrow bRa\)), and transitivity (\(aRb \land bRc \Rightarrow aRc\)). The set \(A\) is obviously infinite, but \(R\) only has two equivalence classes, which are subsets of the integer set \(A\): set of all even integers and set of all odd integers.

Conclusion

Given the counterexample with an infinite set and an equivalence relation that doesn't lead to infinitely many equivalence classes, the initial statement is disproved.

Key concepts

Mathematical Proof

Understanding mathematical proofs is fundamental in verifying and validating mathematical propositions. A mathematical proof is a logical argument demonstrating the truth of a mathematical statement. It starts from known facts and assumptions and, through a series of deductive reasoning steps, arrives at the conclusion.

Proofs can be direct or indirect. A direct proof establishes the truth by a straightforward chain of logical deductions. An indirect proof, such as a proof by contradiction, assumes the opposite of what needs to be proven and shows that this assumption results in a contradiction, thereby proving the original statement to be correct.

When addressing an exercise that asks you to prove or disprove a statement, such as whether an equivalence relation on an infinite set has infinitely many equivalence classes, the first step is to understand what is being asserted. If the intention is to prove the statement, every single step of reasoning must be justified. If the intention is to disprove, one can seek a counterexample that invalidates the statement.

Counterexample

A counterexample is a specific case or instance that shows a general statement to be false. Counterexamples play a crucial role in mathematical proofs, especially when disproving a given proposition.

To use a counterexample effectively, it must be within the constraints set by the original statement but violate the conclusion. For example, in disproving the assertion that an equivalence relation on an infinite set implies infinitely many equivalence classes, a single counterexample of an infinite set with an equivalence relation having only a finite number of equivalence classes is sufficient to disprove the assertion.

Adding to the solution provided, using the relation defined by the difference being an even number on an infinite set of integers, yields exactly two equivalence classes: the evens and the odds. This clear example demonstrates concretely how a single counterexample can overturn a generalized proposition.

Equivalence Classes

An equivalence class is a subset of a set formed by grouping together all elements that are equivalent to each other based on a given equivalence relation. Equivalence relations are relations that are reflexive (each element is related to itself), symmetric (if an element is related to another, then the second is related to the first), and transitive (if an element is related to a second, and the second is related to a third, then the first is related to the third).

In the context of the provided exercise, an equivalence class for an equivalence relation on the integers might group together all integers with a common property, such as having the same remainder when divided by a given number. The infinite set of integers can indeed be partitioned into equivalence classes based on this relation. However, the number of such classes does not have to be infinite; it depends on the specific relation used—illustrating this with the given counterexample where integers grouped by even/odd parity result in two equivalence classes.

An important aspect of equivalence classes is that they partition a set into non-overlapping subsets that cover the entire set. Each element of the set belongs to exactly one equivalence class, embodying the idea that within a class, elements are considered identical in terms of the relation used.