In Class Activity
MAT 300 Spring 2006
April 13, 2006
Recall that a Partition of a set S is a family of subsets, ℑ, that splits S up completely into
separate (non-overlapping) chunks.
Definition: Suppose that S is a set and ℑ = {A} is a family of subsets of S. Then ℑ is a
Partition of S if:
1. ∀A∈ℑ, A≠∅
2. If A, B∈ℑ and A∩B≠∅ then A = B.
3. = S
I
ℑ
A
Recall that an equivalence relation is a relation with the essential properties of equality.
Definition: An equivalence relation, ≡, on a set S is a relation with the following
properties:
1. ∀x∈S, x ≡ x.
2. ∀x, y∈S, x ≡ y ⇒y ≡ x.
3. ∀x, y, z∈S, (x ≡ y and y ≡ z) ⇒x ≡ z.
1. Consider the family of sets consisting of all parabolas of the form y = x2 + a where a
is a real number.
a. Prove that this set of parabolas forms a partition of the plane, ℜ2.
b. Write an equivalence relation that will partition the plane into parabolas of the
form y = x2 + a where a is a real number.
c. Prove that your answer to b. is an equivalence relation.
2. Define the relation, ≡, on ℜ2 as follows: (x,y) ≡ (u,v) ⇔ x2 + y2 = u2 + v2
a. Prove that ≡ is an equivalence relation.
b. Explain how ≡ gives a partition of ℜ2.
c. Describe the partition of ℜ2 given by ≡ and prove that it is a partition.
3. Let S be a set.
a. Explain how an equivalence relation on S will give a partition of S.
b. Prove that an equivalence relation on S will give a partition of S.
4. Let S be a set.
a. Explain how to use a partition, ℑ, of S to define an equivalence relation, ≡, on
S.
b. Prove that ≡ is an equivalence relation on S.