Binary Relations: Understanding Reflexivity, Symmetry, and Transitivity, Study notes of Chinese

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Binary Relations

Problem Set Two checkpoint due in the box up front if you're using a late period. Problem Set Two checkpoint due in the box up front if you're using a late period.

Studying Relationships

● We have just explored the graph as a way of studying relationships between objects. ● However, graphs are not the only formalism we can use to do this.

Binary Relations

● Intuitively speaking: a binary relation over a set A is some relation R where, for every x , y ∈ A , the statement xRy is either true or false. ● Examples: ● < can be a binary relation over ℕ, ℤ, ℝ, etc. ● ↔ can be a binary relation over V for any undirected graph G = ( V , E ). ● (^) ≡ ₖ is a binary relation over ℤ for any integer k. ● We'll give a formal definition later today.

Binary Relations and Graphs

● (^) We can visualize a binary relation R over a set A as a graph: ● (^) The nodes are the elements of A. ● (^) There is an edge from x to y iff xRy. ● Example: the relation a | b (meaning “ a divides b ”) over the set {1, 2, 3, 4} looks like this: 1 2 4 3

Binary Relations and Graphs

● (^) We can visualize a binary relation R over a set A as a graph: ● (^) The nodes are the elements of A. ● (^) There is an edge from x to y iff xRy. ● Example: the relation a = b over {1, 2, 3, 4} looks like this: 1 2 4 3

Categorizing Relations

● Collectively, there are few properties shared by all relations. ● (^) We often categorize relations into different types to study relations with particular properties. ● (^) General outline for today: ● (^) Find certain properties that hold of the relations we've seen so far. ● (^) Categorize relations based on those properties. ● (^) See what those properties entail.

An Intuition for Reflexivity

For every x ∈ A , the relation xRx holds.

Symmetry

● In some relations, the relative order of the objects doesn't matter. ● Examples: ● If x = y , then y = x. ● (^) If u ↔ v , then v ↔ u. ● (^) If x ≡ ₖ y , then y ≡ ₖ x. ● These relations are called symmetric. ● (^) Formally: A binary relation R over a set A is called symmetric iff for all x , y ∈ A , if xRy , then yRx.

Transitivity

● Many relations can be chained together. ● Examples: ● If x = y and y = z , then x = z. ● If u ↔ v and v ↔ w , then u ↔ w. ● (^) If x ≡ ₖ y and y ≡ ₖ z , then x ≡ ₖ z. ● These relations are called transitive. ● Formally: A binary relation R over a set A is called transitive iff for all x , y , z ∈ A , if xRy and yRz , then xRz.

An Intuition for Transitivity

For any x , y , z ∈ A, if xRy and yRz , then xRz.

xRy ≡ x and y have the same shape.

xRy ≡ x and y have the same color.

Closing the Loop

● In any graph G = ( V , E ), we saw that the connected component containing a node v ∈ V is given by { x ∈ V | v ↔ x } ● What is the equivalence class for some node v ∈ V under the relation ↔? [ v ] ↔ = { x ∈ V | v ↔ x } ● Connected components are just equivalence classes of ↔!

Why This Matters

● Developing the right definition for a connected component was challenging. ● Proving every node belonged to exactly one equivalence class was challenging. ● Now that we know about equivalence relations, we get both of these for free! ● If you arrive at the same concept in two or more ways, it is probably significant!