#10. Define f: R ® R defined by f(x) = 4x2. Let A = [0, 4]...
#10. Define f: R ® R defined by f(x) = 4x2. Let A = [0, 4] and B = [-3, 0].
#10(a) Find the sets f(A), f(B), f(A) Ç f(B), and f(A Ç B). Is f(A Ç B) = f(A) Ç f(B)?
#10(b) Find the sets which are inverse images:
f -1(A), f -1 (B), f -1 (A) Ç f -1 (B), and f -1 (A Ç B). Is f -1 (A Ç B) = f -1 (A) Ç f -1 (B)?
#10. Define f: R ® R defined by f(x) = 4x2. Let A = [0, 4] and B = [-3, 0].
#10(a) Find the sets f(A), f(B), f(A) Ç f(B), and f(A Ç B). Is f(A Ç B) = f(A) Ç f(B)?
10(b) Find the sets which are inverse images:
f -1(A), f -1 (B), f -1 (A) Ç f -1 (B), and f -1 (A Ç B). Is f -1 (A Ç B) = f -1 (A) Ç f -1 (B)?
4. Find an example of a relation on a set S that
is symmetric , but not reflexive and not transitive. (Note that you
can just specify your set S and list an appropriate set of ordered
pairs.)
#6. State the range of each function f: R ® R. (no work required to be shown)
#6(a) f(x) = 5 sin (4x)
#6(b) f(x) = −(x − 3)2 + 8
Homework Answers
QI. Let A-(-4-3-2-1,0,1,2,3,4]. R İs defined on A as follows: For all (m, n) E A,...
QI. Let A-(-4-3-2-1,0,1,2,3,4]. R İs defined on A as follows: For all (m, n) E A, mRn㈠4](rn2_n2) Show that the relation R is an equivalence relation on the set A by drawing the graph of relation Find the distinct equivalence classes of R. Q2. Find examples of relations with the following properties a) Reflexive, but not symmetric and not transitive. b) Symmetric, but not reflexive and not transitive. c) Transitive, but not reflexive and not symmetric. d) Reflexive and symmetric,...
Question 2 For each of the following relations R, determine (and explain) whether R is: (1) reflexive (2) symmetric (3) antisymmetric (4) transitive (a) R-(x, y):x +2y 3), defined on the set A 10...
Question 2 For each of the following relations R, determine (and explain) whether R is: (1) reflexive (2) symmetric (3) antisymmetric (4) transitive (a) R-(x, y):x +2y 3), defined on the set A 10, 1,2,3) (b) R-I(x, y): xy 4), defined on the set A (0,1,2,3,4 (c) R-(x, y): xy 4), defined on the set A-0,,2,3)
Question 2 For each of the following relations R, determine (and explain) whether R is: (1) reflexive (2) symmetric (3) antisymmetric (4) transitive (a)...
9. Define R the binary relation on N x N to mean (a, b)R(c, d) iff...
9. Define R the binary relation on N x N to mean (a, b)R(c, d) iff b|d and alc (a) R is symmetric but not reflexive. (b) R is transitive and symmetric but not reflexive (c) R is reflexive and transitive but not symmetric (d) None of the above 10. Let R be an equivalence relation on a nonempty and finite
9. Define R the binary relation on N x N to mean (a, b)R(c, d) iff b|d and alc...
Let R be the relation defined on Z (integers): a R b iff a + b...
Let R be the relation defined on Z (integers): a R b iff a + b is even. R is an equivalence relation since R is: Group of answer choices Reflexive, Symmetric and Transitive Symmetric and Reflexive or Transitive Reflexive or Transitive Symmetric and Transitive None of the above
Let R be the relation defined on Z (integers): a R b iff a + b...
Let R be the relation defined on Z (integers): a R b iff a + b is even. Suppose that 'even' is replaced by 'odd' . Which of the properties reflexive, symmetric and transitive does R possess? Group of answer choices Reflexive, Symmetric and Transitive Symmetric Symmetric and Reflexive Symmetric and Transitive None of the above
4. Let S be the set of continuous function f: [0;1) ! R. Let R be...
4. Let S be the set of continuous function f: [0;1) ! R. Let R be the relation defined on S by (f; g) 2 Rif(x) is O(g(x)). (a) Is R reflexive? (b) Is R antisymmetric? (c) is R symmetric? (d) is R transitive? Explain your answer in details. Use the definition of big-O to justify your answer if you think R has a certain property or give a counter example if you think R does not have a certain...
Let R be a relation defined on the integers Z by a R b if 6b^3 - 6a^3 <= 0 Which of the properties reflexive, symmetric, and transitive does R possess?
Let R be a relation defined on the integers Z by a R b if 6b^3 - 6a^3 <= 0 Which of the properties reflexive, symmetric, and transitive does R possess?
(17) (20pt) Let F be the set of functions f : R+ → R. Prove that the binary relation "f is 0(g)" on F is: (a) (4pt) Write down the definition for "f is O(g)". (b) (4pt) Prove that the...
(17) (20pt) Let F be the set of functions f : R+ → R. Prove that the binary relation "f is 0(g)" on F is: (a) (4pt) Write down the definition for "f is O(g)". (b) (4pt) Prove that the relation is reflexive (c) (6pt) Prove that the relation is not symmetric. (d) (6pt) Prove that the relation is transitive.
(17) (20pt) Let F be the set of functions f : R+ → R. Prove that the binary relation "f...
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?
4. Define a function f:N → Z by tof n/2 if n is even 1-(n +...
4. Define a function f:N → Z by tof n/2 if n is even 1-(n + 1)/2 if n is odd. f(n) = Show that f is a bijection. 11 ] 7. Let X = R XR and let R be a relation on X defined as follows ((x,y),(w,z)) ER 4 IC ER\ {0} (w = cx and z = cy.) Is R reflexive? Symmetric? Transitive? An equivalence relation? Explain each of your answers. Describe the equivalence classes [(0,0)]R and...
QI. Let A-(-4-3-2-1,0,1,2,3,4]. R İs defined on A as follows: For all (m, n) E A, mRn㈠4](rn2_n2) Show that the relation R is an equivalence relation on the set A by drawing the graph of relation Find the distinct equivalence classes of R. Q2. Find examples of relations with the following properties a) Reflexive, but not symmetric and not transitive. b) Symmetric, but not reflexive and not transitive. c) Transitive, but not reflexive and not symmetric. d) Reflexive and symmetric,...
Question 2 For each of the following relations R, determine (and explain) whether R is: (1) reflexive (2) symmetric (3) antisymmetric (4) transitive (a) R-(x, y):x +2y 3), defined on the set A 10, 1,2,3) (b) R-I(x, y): xy 4), defined on the set A (0,1,2,3,4 (c) R-(x, y): xy 4), defined on the set A-0,,2,3)
Question 2 For each of the following relations R, determine (and explain) whether R is: (1) reflexive (2) symmetric (3) antisymmetric (4) transitive (a)...
9. Define R the binary relation on N x N to mean (a, b)R(c, d) iff b|d and alc (a) R is symmetric but not reflexive. (b) R is transitive and symmetric but not reflexive (c) R is reflexive and transitive but not symmetric (d) None of the above 10. Let R be an equivalence relation on a nonempty and finite
9. Define R the binary relation on N x N to mean (a, b)R(c, d) iff b|d and alc...
4. Let S be the set of continuous function f: [0;1) ! R. Let R be the relation defined on S by (f; g) 2 Rif(x) is O(g(x)). (a) Is R reflexive? (b) Is R antisymmetric? (c) is R symmetric? (d) is R transitive? Explain your answer in details. Use the definition of big-O to justify your answer if you think R has a certain property or give a counter example if you think R does not have a certain...
(17) (20pt) Let F be the set of functions f : R+ → R. Prove that the binary relation "f is 0(g)" on F is: (a) (4pt) Write down the definition for "f is O(g)". (b) (4pt) Prove that the relation is reflexive (c) (6pt) Prove that the relation is not symmetric. (d) (6pt) Prove that the relation is transitive.
(17) (20pt) Let F be the set of functions f : R+ → R. Prove that the binary relation "f...
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?
4. Define a function f:N → Z by tof n/2 if n is even 1-(n + 1)/2 if n is odd. f(n) = Show that f is a bijection. 11 ] 7. Let X = R XR and let R be a relation on X defined as follows ((x,y),(w,z)) ER 4 IC ER\ {0} (w = cx and z = cy.) Is R reflexive? Symmetric? Transitive? An equivalence relation? Explain each of your answers. Describe the equivalence classes [(0,0)]R and...
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