Practice Equivalence Classes in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
An equivalence class is the set of all elements that are related to a given element under an equivalence relation — it groups objects that are considered 'the same' in some specified sense.
Treating different things as equal because they share what matters.
Example 1
mediumDefine the relation a \sim b on \mathbb{Z} by 'a \sim b iff a \equiv b \pmod{3}'. Verify this is an equivalence relation and list the equivalence classes.
Example 2
mediumOn the set of triangles, define T_1 \sim T_2 if T_1 is congruent to T_2. Show this is an equivalence relation.
Example 3
easyList the equivalence classes of \mathbb{Z} under a \sim b iff a \equiv b \pmod{2}.
Example 4
mediumDefine f \sim g on functions from \mathbb{R} to \mathbb{R} by f \sim g iff f(0) = g(0). Verify this is an equivalence relation and describe the equivalence class of f(x) = x^2.