week2 Relations

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Define Cartesian product
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a Cartesian product is a mathematical operation that returns a set from multiple sets

What is the Cartesian product of two sets: A = {a, b} and B = {5, 6}.
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defining two sets: A = {a, b} and B = {5, 6}. Both set A and set B consist of two elements each. Their Cartesian product, written as A × B, results in a new set which has the following elements: A × B = {(a,5), (a,6), (b,5), (b,6)}.

Define from to relations
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LetAandBbe nonempty sets. A (binary) relationTfromAtoBisa subset ofA B. IfT A Band(a,b)2T, we say thatais related tobbyT, aTb

What is the more common way of describing relations?
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\by characteristics of their elements"

give an example of a relationship defined by charicteristics
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x ∈ A is related to y ∈ B if and only if x<=y

What is a relation on a set
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When A=B then a (binary) relation on A is a relation from A to A, hence a subset of A * A

Explain the property reflexiveity
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R is reflexive if and only if(x,x) ∈ R for all x∈ A.

Explain the property symmetry
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R is symmetric if and only if for all x, y ∈A if (x,y) ∈ R then(y,x)∈R

Explain the property transitivity
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Ris transitive if and only if for all x, y, z ∈ A if(x,y) ∈ R and(y,z) ∈ R then(x,z) ∈ R

Explain the property equivalence
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Ris an equivalence relation if and only ifRis re exive, symmetric, and transitive

Explain Equivalence class
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Suppose A is a set and R is an equivalence relation on A. For each elementain A, the equivalence class of a,[a], is the set of allelementxinAsuch thatxis related toabyR:[a] =fxjx2Aand(x,a)2Rg

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