Let S = 1, 2, 3 and ρ = {(1, 3),(3, 3),(2, 2),(2, 3),(1, 1),(1, 2)},...
Let S = 1, 2, 3 and ρ = {(1, 3),(3, 3),(2, 2),(2, 3),(1, 1),(1, 2)}, a binary relation on S.
(a) Test ρ for reflexivity, symmetry, antisymmetry and transitivity.
(b) Find ρ ∗ , the closure of the binary relation under symmetry. If ρ is already symmetric, then state so.
Homework Answers
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Let S = 1, 2, 3 and ρ = {(1, 3),(3, 3),(2, 2),(2, 3),(1, 1),(1,
2)},...
I. Let each of R, S, and T be binary relations on N2 as defined here: R-[<m, n EN nis the smallest prime number greater than or equal to m] S -[< m, n> EN* nis the greatest prime number less...
I. Let each of R, S, and T be binary relations on N2 as defined here: R-[<m, n EN nis the smallest prime number greater than or equal to m] S -[< m, n> EN* nis the greatest prime number less than or equal to m] (a) Which (if any) of these binary relations is a (unary) function? (b) Which (if any) of these binary relations is an injection? (c) Which (if any) of these binary relations is a surjection?...
6. Let S-11, 2, 3, 6, 8, 10). For x,yeS, let x S y if xly. Answer the questions below: a) Is this an Equivalence Relation? Remember to check all three criteria (Reflexivity, Symmetry, and Transi...
6. Let S-11, 2, 3, 6, 8, 10). For x,yeS, let x S y if xly. Answer the questions below: a) Is this an Equivalence Relation? Remember to check all three criteria (Reflexivity, Symmetry, and Transitivity). Be sure to give a short explanatio if the property holds and a specific counterexample if it does not hold. b) Is this a Partial Ordering? Remember to check all three criteria (Reflexivity Transitivity, and Anti-Symmetry). Be sure to give a short explanation if...
56. Let S = N × N and let ρ be a binary relation on şdefined...
56. Let S = N × N and let ρ be a binary relation on şdefined by (x,y)ρ(z, w)艹x + y-z + w. Show that p is an equivalence relation on S and describe the resulting equivalence classes.
5. (6 marks) Let S be the set of all binary strings of length 6. Consider the relation ρ on the set S in which for all a,b ∈ S, (a,b) ∈ ρ if and only if the length of a longest substring of consecutiv...
5. (6 marks) Let S be the set of all binary strings of length 6. Consider the relation ρ on the set S in which for all a,b ∈ S, (a,b) ∈ ρ if and only if the length of a longest substring of consecutive ones in a is the same as the length of a longest substring of consecutive ones in b. (a) Is 011010 related to 000011? Explain why or why not. (b) Prove that ρ is an...
4. [3 marks] Let R be a relation on a set A. Let A {1,2, 3,...
4. [3 marks] Let R be a relation on a set A. Let A {1,2, 3, X, Y} and R = {(1, 1), (1,3), (2,1), (3, 1), (1, X), (X, Y)} (a) What is the reflexive closure of R? (b) What is the symmetric closure of R? (c) What is the transitive closure of R?
(a) Let S be a symmetric positive definite matrix and define a function | on R" by 1/2 xx Sx . Pr...
(a) Let S be a symmetric positive definite matrix and define a function | on R" by 1/2 xx Sx . Prove that this function defines a vector norm. Hint: Use the Cholesky decomposition. (b) Find an example of square matrices A an This shows that ρ(A) is not a norm. Note: there are very simple examples. d B such that ρ(A+B)>ρ(A) + ρ(8)
(a) Let S be a symmetric positive definite matrix and define a function | on R"...
For part (a), please prove the answer. 5. Let S = {1, 2, 3, 4} and...
For part (a), please prove the answer.
5. Let S = {1, 2, 3, 4} and let F be the sets of all functions from S to S. Let R be the relation on F defined by: For all f,g EF, fRg if and only if fog(1)-2. (a) Is R reflexive? symmetric? transitive? (b) Is it true that that there exists f E F so that fRf? Prove your answer. (c) Is it true that for all f F, there...
(4) (a) Give an example of a relation (different to those in question 1) which is...
(4) (a) Give an example of a relation (different to those in question 1) which is symmetric and transitive but not reflexive. (b) Identify the problem with the following proof: Let R be a relation on a set S, and suppose that R is symmetric and transitive. Since the relation is symmetric, we know that a bb~a, and then it follows from transitivity that a ~b and b ~ a → a ~ a. Therefore any symmetric and transitive relation...
2. Let S 11,2,3,4,5, 6, 7,8,91 and let T 12,4,6,8. Let R be the relation on P (S) detined by for ...
2. Let S 11,2,3,4,5, 6, 7,8,91 and let T 12,4,6,8. Let R be the relation on P (S) detined by for all X, Y E P (s), (X, Y) E R if and only if IX-T] = IY-T]. (a) Prove that R is an equivalence relation. (b) How many equivalence classes are there? Explain. (c) How mauy elements of [ø], the equivalence class of ø, are there? Explain (d) How many elements of [f1,2,3, 4)], the equivalence class of (1,2,3,...
Let R ⊆ {1, 2, 3, 4} × {1, 2, 3, 4} be the relation R...
Let R ⊆ {1, 2, 3, 4} × {1, 2, 3, 4} be the relation R = {(1,
3),(1, 4),(2, 2),(2, 4),(3, 1),(3, 2),(4, 4)}. (a) Compute R −1 .
(b) Compute the relations R ∪ R −1 and R ∩ R −1 , and check that
they are symmetric.
7.1.3 Let RC 1,2,3,4) x 1,2,3,4) be the relation (a) Compute R-1 (b) Compute the relations RUR-1 and RnR-1, and check that they are symmetric.
I. Let each of R, S, and T be binary relations on N2 as defined here: R-[<m, n EN nis the smallest prime number greater than or equal to m] S -[< m, n> EN* nis the greatest prime number less than or equal to m] (a) Which (if any) of these binary relations is a (unary) function? (b) Which (if any) of these binary relations is an injection? (c) Which (if any) of these binary relations is a surjection?...
6. Let S-11, 2, 3, 6, 8, 10). For x,yeS, let x S y if xly. Answer the questions below: a) Is this an Equivalence Relation? Remember to check all three criteria (Reflexivity, Symmetry, and Transitivity). Be sure to give a short explanatio if the property holds and a specific counterexample if it does not hold. b) Is this a Partial Ordering? Remember to check all three criteria (Reflexivity Transitivity, and Anti-Symmetry). Be sure to give a short explanation if...
56. Let S = N × N and let ρ be a binary relation on şdefined by (x,y)ρ(z, w)艹x + y-z + w. Show that p is an equivalence relation on S and describe the resulting equivalence classes.
4. [3 marks] Let R be a relation on a set A. Let A {1,2, 3, X, Y} and R = {(1, 1), (1,3), (2,1), (3, 1), (1, X), (X, Y)} (a) What is the reflexive closure of R? (b) What is the symmetric closure of R? (c) What is the transitive closure of R?
(a) Let S be a symmetric positive definite matrix and define a function | on R" by 1/2 xx Sx . Prove that this function defines a vector norm. Hint: Use the Cholesky decomposition. (b) Find an example of square matrices A an This shows that ρ(A) is not a norm. Note: there are very simple examples. d B such that ρ(A+B)>ρ(A) + ρ(8)
(a) Let S be a symmetric positive definite matrix and define a function | on R"...
For part (a), please prove the answer.
5. Let S = {1, 2, 3, 4} and let F be the sets of all functions from S to S. Let R be the relation on F defined by: For all f,g EF, fRg if and only if fog(1)-2. (a) Is R reflexive? symmetric? transitive? (b) Is it true that that there exists f E F so that fRf? Prove your answer. (c) Is it true that for all f F, there...
(4) (a) Give an example of a relation (different to those in question 1) which is symmetric and transitive but not reflexive. (b) Identify the problem with the following proof: Let R be a relation on a set S, and suppose that R is symmetric and transitive. Since the relation is symmetric, we know that a bb~a, and then it follows from transitivity that a ~b and b ~ a → a ~ a. Therefore any symmetric and transitive relation...
2. Let S 11,2,3,4,5, 6, 7,8,91 and let T 12,4,6,8. Let R be the relation on P (S) detined by for all X, Y E P (s), (X, Y) E R if and only if IX-T] = IY-T]. (a) Prove that R is an equivalence relation. (b) How many equivalence classes are there? Explain. (c) How mauy elements of [ø], the equivalence class of ø, are there? Explain (d) How many elements of [f1,2,3, 4)], the equivalence class of (1,2,3,...
Let R ⊆ {1, 2, 3, 4} × {1, 2, 3, 4} be the relation R = {(1,
3),(1, 4),(2, 2),(2, 4),(3, 1),(3, 2),(4, 4)}. (a) Compute R −1 .
(b) Compute the relations R ∪ R −1 and R ∩ R −1 , and check that
they are symmetric.
7.1.3 Let RC 1,2,3,4) x 1,2,3,4) be the relation (a) Compute R-1 (b) Compute the relations RUR-1 and RnR-1, and check that they are symmetric.
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