Equivalence class & partition of sets

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This discrete mathematics lecture introduces equivalence classes and set partitions using a worked example. The instructor presents the set A = {1, 2, 3, 4, 5} and an equivalence relation R = {(1,1),(2,2),(3,3),(4,4),(5,5),(1,4),(4,1),(2,5),(5,2)}, asking students to find the partition of A defined by R. The non-reflexive pairs (1,4), (4,1), (2,5), and (5,2) are highlighted to identify which elements are related. The equivalence class of an element x is defined as [x] = {y | y ∈ A and (x, y) ∈ R}. Applying this definition yields [1] = {1, 4}, [2] = {2, 5}, [3] = {3}, [4] = {1, 4}, and [5] = {2, 5}. The instructor notes that [x] can equal [y] even when x ≠ y, so the partition consists of the distinct classes {1, 4}, {2, 5}, and {3}. A partition is then defined as a subdivision of a set into non-empty, non-overlapping subsets whose union equals the original set and whose intersection is empty.

Chapters

1. 0:00 – 2:00 00:00-02:00

   The problem is displayed on a dark board: “Consider A = {1, 2, 3, 4, 5} an equivalence relation R on A” with R = {(1,1),(2,2),(3,3),(4,4),(5,5),(1,4),(4,1),(2,5),(5,2)} and the instruction “find the partition of a set A, defined by R.” The instructor points to the ordered pairs (1,4), (4,1), (2,5), and (5,2), which are underlined in green as a teaching cue to identify related elements. A vertical list of prompts “[1] =”, “[2] =”, “[3] =”, “[4] =”, and “[5] =” is shown, and the instructor begins writing solutions such as [1] = {1, 4} and [2] = {2, 5}.

2. 2:00 – 4:05 02:00-04:05

   The completed board work shows [1] = {1, 4}, [2] = {2, 5}, [3] = {3}, [4] = {1, 4}, and [5] = {2, 5}. A definition slide states “Equivalence Class: - of an element is denoted by [x]” and gives “[x] = {y | y ∈ A and (x, y) ∈ R} for all x ∈ A,” with the note “We can have [x] = [y], even if x != y.” The instructor then presents “Partitions of a Set,” displaying the conditions A1 U A2 U ... U An = A and A1 ∩ A2 ∩ ... ∩ An = Φ, explaining that a partition is a subdivision into non-empty, non-overlapping subsets. The distinct equivalence classes {1, 4}, {2, 5}, and {3} are summarized as the partition of A.

The lesson progresses from a concrete problem to general definitions. First, the equivalence relation R on A = {1, 2, 3, 4, 5} is analyzed by identifying the non-reflexive pairs that connect elements: (1,4) and (4,1) link 1 and 4, while (2,5) and (5,2) link 2 and 5. Using the definition [x] = {y | y ∈ A and (x, y) ∈ R}, each equivalence class is computed. The key insight is that equivalent elements share the same class, so [1] = [4] and [2] = [5], reducing the five classes to three distinct subsets. The partition is then formally defined by two conditions: the union of all parts equals A, and their intersection is empty (Φ). This connects equivalence relations to partitions: every equivalence relation induces a unique partition of the set into its distinct equivalence classes.