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 rRy if and only if f(x)Sf(y), for all I, YE 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.
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 rRy if and only if f(x)Sf(y), for all I, YE 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.
Advanced Engineering Mathematics
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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I need help proving only problem 5 for this practice exercise. I need the full proof with the definition.
![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 E 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
E A. Choose y A such that rRy. 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, QE P(A) are related if and only if PC 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 UzEA[x] = A.
2. Suppose R is symmetric. Prove that x E [y] if and only if y € [x], for all x, y € A.
3. Suppose R is transitive. Prove that if Ry, 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 zy if and only if f(x) = f(y), for all x, y E A. Prove that is an equivalence
relation.
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Transcribed Image Text: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 E 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
E A. Choose y A such that rRy. 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, QE P(A) are related if and only if PC 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 UzEA[x] = A.
2. Suppose R is symmetric. Prove that x E [y] if and only if y € [x], for all x, y € A.
3. Suppose R is transitive. Prove that if Ry, 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 zy if and only if f(x) = f(y), for all x, y E A. Prove that is an equivalence
relation.
1
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