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Problems and Theorems in Set Theory and Relations, Assignments of Abstract Algebra

The University of Tennessee (UT)Abstract Algebra

A series of problems and theorems related to set theory and relations, including proving set equalities, finding domains and ranges of relations, and proving properties of relations such as reflexive, symmetric, and transitive. It also covers equivalence relations and their properties.

Typology: Assignments

Pre 2010

Uploaded on 08/27/2009

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Math 300 – Problems #6 - COMPLETE
In the exercises that follow A,B,Cand Dare arbitrary sets, unless described otherwise.
167. If A= (−∞,0],B= [0,∞), and C= (0,1), find and draw each of the following sets: A×(B∩
C),(A∩B)×C.
168. If A= (−∞,0],B= [0,∞), and C= (0,1), find and draw each of the following sets: (A\B)×
(A∩B),(A×B)∩(B×C).
169. Prove if A⊆Band C⊆Dthen A×C⊆B×D.
170. Prove that A×(B∩C) = (A×B)∩(A×C).
171. Prove that A× ∅ =∅.
172. Prove that (A∩B)×(C∩D) = (A×C)∩(B×D).
173. Prove that (A∪B)×C= (A×C)∪(B×C).
174. Prove or disprove that (A∪B)×(C∪D) = (A×C)∪(B×D).
175. Prove that A×(B\C) = (A×B)\(A×C).
176. Let B⊆Aand prove that (A×A)\(B×B) = ((A\B)×A)×(A×(A\B)).
177. Let C6=∅and prove that if A×C=B×Cthen A=B.
178. Let A6=Band prove that if A×C=B×Cthen C=∅.
179. Find the domain and range of the following relation: xRymeans x, y ∈Rand y= 3 ·(x−1)2+ 2.
180. Find the domain and range of the following relation: xRymeans x, y ∈Rand y= 2 ·x2−8·x+ 9.
181. Find the domain and range of the following relation: xRymeans x, y ∈Rand x2−y2= 1.
182. Find the domain and range of the following relation: xRymeans x, y ∈Rand xy =x2−1.
183. Find the domain and range of the following relation: xRymeans x, y ∈Rand √1−x= (x+ 1)y.
184. For each of the following relations, prove or find a counterexample for each of the properties: reflexive,
symmetric and transitive.
(a) Define xRyon Z×Zto mean x−yis a nonnegative integer.
(b) Define xRyon N×Nto mean xy is an integer.
(c) Define xRyon Z×Zto mean x−yis an odd integer.
(d) Define xRyon R×Rto mean x−yis irrational.
(e) Define xRyon R×Rto mean x≥y.
185. For each of the following relations prove that it is an equivalence relation.
(a) Given x, y ∈[0,∞),xRymeans y=√x2.
(b) Given ordered pairs (a, b),(p, q)∈R×R,(a, b)R(p, q)means pb =aq.
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Math 300 – Problems #6 - COMPLETE

In the exercises that follow A, B, C and D are arbitrary sets, unless described otherwise.

  1. If A = (−∞, 0], B = [0, ∞), and C = (0, 1), find and draw each of the following sets: A × (B ∩ C), (A ∩ B) × C.
  2. If A = (−∞, 0], B = [0, ∞), and C = (0, 1), find and draw each of the following sets: (A \ B) × (A ∩ B), (A × B) ∩ (B × C).
  3. Prove if A ⊆ B and C ⊆ D then A × C ⊆ B × D.
  4. Prove that A × (B ∩ C) = (A × B) ∩ (A × C).
  5. Prove that A × ∅ = ∅.
  6. Prove that (A ∩ B) × (C ∩ D) = (A × C) ∩ (B × D).
  7. Prove that (A ∪ B) × C = (A × C) ∪ (B × C).
  8. Prove or disprove that (A ∪ B) × (C ∪ D) = (A × C) ∪ (B × D).
  9. Prove that A × (B \ C) = (A × B) \ (A × C).
  10. Let B ⊆ A and prove that (A × A) \ (B × B) = ((A \ B) × A) × (A × (A \ B)).
  11. Let C 6 = ∅ and prove that if A × C = B × C then A = B.
  12. Let A 6 = B and prove that if A × C = B × C then C = ∅.
  13. Find the domain and range of the following relation: x R y means x, y ∈ R and y = 3 · (x − 1)^2 + 2.
  14. Find the domain and range of the following relation: x R y means x, y ∈ R and y = 2 · x^2 − 8 · x + 9.
  15. Find the domain and range of the following relation: x R y means x, y ∈ R and x^2 − y^2 = 1.
  16. Find the domain and range of the following relation: x R y means x, y ∈ R and xy = x^2 − 1.
  17. Find the domain and range of the following relation: x R y means x, y ∈ R and √ 1 − x = (x + 1)y.
  18. For each of the following relations, prove or find a counterexample for each of the properties: reflexive, symmetric and transitive. (a) Define x R y on Z × Z to mean x − y is a nonnegative integer. (b) Define x R y on N × N to mean xy is an integer. (c) Define x R y on Z × Z to mean x − y is an odd integer. (d) Define x R y on R × R to mean x − y is irrational. (e) Define x R y on R × R to mean x ≥ y.
  19. For each of the following relations prove that it is an equivalence relation. (a) Given x, y ∈ [0, ∞), x R y means y = √x^2. (b) Given ordered pairs (a, b), (p, q) ∈ R × R, (a, b) R (p, q) means pb = aq.

(c) Given x, y ∈ R, x R y means x − y is an integer. (d) Given x, y ∈ R \ { 0 }, x R y means x/y = ± 1. (e) Let n be a positive natural number. Define a relation R on the set of integers Z by: x R y if and only if x − y is divisible by n. (f) Define a relation R on the set of integers Z as follows: x R y if and only if x + y is even. (g) Define a relation R on the set of reals as follows: x R y if and only if x − y is rational. (h) A set X is the union of 6 subsets A 1 ,... , A 6 which are mutually disjoint (that means Ai ∩Aj = ∅ if i 6 = j). Define the relation R on X as follows: x R y if and only if there is i so that Ai contains both x and y.

  1. Let R be a relation on X × Y. Given x ∈ X we define the subset Ax of Y as follows: Ax = {y ∈ Y | x R y}. Prove that if R is an equivalence relation (and X = Y ) then for each pair x, y ∈ X then either Ax = Ay or Ax ∩ Ay = ∅ (identical or disjoint).
  2. A set X contains 9 subsets A 1 ,... , A 9 not necessarily disjoint. We define the relation R on X as follows: x R y if and only if there is i so that Ai contains both x and y. Prove that R is a reflexive relation if and only if X is the union of all of Ai, i ≤ 9.
  3. A set X contains 9 subsets A 1 ,... , A 9. None of them is contained in the union of remaining ones. Define the relation R on X as follows: x R y if and only if there is i so that Ai contains both x and y. Prove that R is a transitive relation if all of Ai, i ≤ 9 are mutually disjoint.
  4. Suppose R is a symmetric relation on X × X which satisfies

Domain ( R ) ∪ Range ( R ) = X. Prove that Domain ( R ) = X = Range ( R ).

  1. Suppose R is a symmetric and transitive relation on X × X which satisfies

Domain ( R ) ∪ Range ( R ) = X. Prove that R is an equivalence relation on X.

  1. Suppose R and S are equivalence relations on X. Prove that R ∩ S is an equivalence relation on X. What about R ∪ S?
  2. Suppose R is a reflexive relation on X × X with domain X which satisfies the following condition: if x R y and x R z, then y R z. Prove that R is an equivalence relation on X. Does every equivalence relation satisfy this condition?