Problem 1 Let T be the set of all triangles in the plane, and let R be the relation on T defined by sRt if and only if s has greater area than t for all triangles s, t ∈T . Is this relation reflexive, symmetric, and/or transitive?

Question

Fall 2022 Math 300: Exercise # 11
Problem 1
Let T be the set of all triangles in the plane, and let R be the relation on T defined by
sRt if and only if s has greater area than t for all triangles s, t ∈T . Is this relation reflexive,
symmetric, and/or transitive?

Problem 2
Let A be a set, and let R be a relation on A. Suppose that R is symmetric and transitive.
Find the flaw in the following alleged proof that this relation is necessarily reflexive. “Let
x ∈A. Choose y ∈A such that xRy. By symmetry know that yRx, and then by transitivity
we see that xRx. Hence R is reflexive.”

Problem 3
The power set of any set A is denoted by P (A) and is defined as the set of all subsets of
A, including the empty set and the set A itself. Think of ⊆ as defining a relation on P (A)
where P, Q ∈ P (A) are related if and only if P ⊆ Q. Is this relation reflexive, symmetric,
and/or transitive?

Problem 4
Let A be a set, and let R be a relation on A.
1. Suppose R is reflexive. Prove that ∪x∈A[x] = A.
2. Suppose R is symmetric. Prove that x ∈[y] if and only if y ∈[x], for all x, y ∈A.
3. Suppose R is transitive. Prove that if xRy, then [y] ⊆[x] for all x, y ∈A.

Problem 5
Let A and B be sets, let R and S be relations on A and B, respectively, and let f : A →B
be a function. The function f is relation preserving if xRy if and only if f (x)Sf (y), for all
x, y ∈A.
1. Suppose that f is bijective & relation preserving. Prove that f^(−1) is relation preserving.
2. Suppose that f is surjective & relation preserving. Prove that R is reflexive, symmetric
or transitive if and only if S is reflexive, symmetric or transitive, respectively.

Problem 6
Let A and B be sets, and let f : A → B be a function. Let ∼ be the relation on A
defined by x ∼ y if and only if f (x) = f (y), for all x, y ∈A. Prove that ∼ is an equivalence
relation.

Problem 7
Let A be a set, and let ≈ be a relation on A. Prove that is an equivalence relation if
and only if the following two conditions hold.
1. x ≈x for all x ∈A.
2. x ≈y and y ≈z implies z ≈x, for all x, y, z ∈A.
Problem 8
Let F be the family of subsets of [0, ∞) defined by
F = {[2^(n−1) −1, 2^(n)-1)} n∈N.
Prove that this family of sets is a partition of [0, ∞).
Problem 9
For the following equivalence relation describe the corresponding partition without any
redundancy or reference to the name of the relation. Let ∼be the relation on R−{0}defined
by x ∼y if and only if xy > 0, for all x, y ∈R−{0}.
Problem 10
For the following partition describe the corresponding equivalence relation without any
reference to the name of the partition. Let D be the partition of R2 consisting of all circles
in R2 centered at the origin (the origin itself is considered a “degenerate” circle).