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ng R is symmetric and transitive but not reflexive. 14. Let A be a set with at least three elements. abivib lo so (a) If P = {B1, B2} is a partition of A with B, + B,, is {B{, B5} a partition 1 ei of A? Explain. What if B, = B,?gon ovdoie (b) If P = {B1, B2, B3} is a partition of A, is {B{, B5, B§} a partition of A? Explain. Consider the possibility that two or more of the elements of P may be equal.omdi (c) If P = {B1, B2} is a partition of A, C, is a partition of B1, C2 is a parti- coubnanon oolo bolleo dgim luqmow denw g tion of B, and B¡ + B,, prove that C, U C, is a partition of A. bort

ng R is symmetric and transitive but not reflexive. 14. Let A be a set with at least three elements. abivib lo so (a) If P = {B1, B2} is a partition of A with B, + B,, is {B{, B5} a partition 1 ei of A? Explain. What if B, = B,?gon ovdoie (b) If P = {B1, B2, B3} is a partition of A, is {B{, B5, B§} a partition of A? Explain. Consider the possibility that two or more of the elements of P may be equal.omdi (c) If P = {B1, B2} is a partition of A, C, is a partition of B1, C2 is a parti- coubnanon oolo bolleo dgim luqmow denw g tion of B, and B¡ + B,, prove that C, U C, is a partition of A. bort

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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please help me on question 14

ve and symmetric but not tran- {yE A: x R y}. (Note that R(x) is the same as x except (A) = that R is not an equivalence relation in this exercise.) Does the set A = {R(x): x E A } always form a partition of A? Prove that your answer is correct. 12. Repeat Exercise 11, assuming R is reflexive and transitive but not symmetric. no nou. 13. Repeat Exercise 11, assuming R is symmetric and transitive but not reflexive. vib ai 14. Let A be a set with at least three elements. (a) If P = of A? Explain. What if B, = B,?g ow doidw (b) If P = {B1, B2, B3} is a partition of A, is {Bº, B§, B§} a partition of A? Explain. Consider the possibility that two or more of the elements of P may be equal. (c) If P = {B1, B2} is a partition of A, 6, is a partition of B1, C2 is a parti- {B1, B2} is a partition of A with B, + B,, is {B“, B5} a partition GG C %3D 29 %3D Combne FradeDo 15. Assign a grade of A (correct), C (partially correct), or F (failure) to each. slugmoo w dsnWgtion of B,, and B # B,, prove that C, U6, is a partition of A. 1 yodi Jud oot Justify assignments of grades other than A. (GLGUCG (a) Claim. Let R be an equivalence relation on the set A, and let x, y, and z be elements of A. If x E y and z ¢ x, then z ¢ ỹ. un diw oilomdiis ol, In i noqu bnoqob Jeri «Proof.9 101 210w oolo nuitivity, v Rz. so z E v. Therefore, if xe y and z x, then z & y. E 1odi yea biu Assume that xe ỹ and z e x. Then y R x and x R z. By tran- Let R be an equivalence relation on the set A, and let x, y, and (b) Claim. (o bom) z be elements of A. If x e ỹ and z ¢ x, then z ¢ ỹ. "Proof." Assume that xE y, and assume that z E y. Then y R x and 99 y R z. By symmetry, x R y, and by transitivity, x R z. Therefore, z e I. and z & X. then z ¢ y.
Transcribed Image Text:ve and symmetric but not tran- {yE A: x R y}. (Note that R(x) is the same as x except (A) = that R is not an equivalence relation in this exercise.) Does the set A = {R(x): x E A } always form a partition of A? Prove that your answer is correct. 12. Repeat Exercise 11, assuming R is reflexive and transitive but not symmetric. no nou. 13. Repeat Exercise 11, assuming R is symmetric and transitive but not reflexive. vib ai 14. Let A be a set with at least three elements. (a) If P = of A? Explain. What if B, = B,?g ow doidw (b) If P = {B1, B2, B3} is a partition of A, is {Bº, B§, B§} a partition of A? Explain. Consider the possibility that two or more of the elements of P may be equal. (c) If P = {B1, B2} is a partition of A, 6, is a partition of B1, C2 is a parti- {B1, B2} is a partition of A with B, + B,, is {B“, B5} a partition GG C %3D 29 %3D Combne FradeDo 15. Assign a grade of A (correct), C (partially correct), or F (failure) to each. slugmoo w dsnWgtion of B,, and B # B,, prove that C, U6, is a partition of A. 1 yodi Jud oot Justify assignments of grades other than A. (GLGUCG (a) Claim. Let R be an equivalence relation on the set A, and let x, y, and z be elements of A. If x E y and z ¢ x, then z ¢ ỹ. un diw oilomdiis ol, In i noqu bnoqob Jeri «Proof.9 101 210w oolo nuitivity, v Rz. so z E v. Therefore, if xe y and z x, then z & y. E 1odi yea biu Assume that xe ỹ and z e x. Then y R x and x R z. By tran- Let R be an equivalence relation on the set A, and let x, y, and (b) Claim. (o bom) z be elements of A. If x e ỹ and z ¢ x, then z ¢ ỹ. "Proof." Assume that xE y, and assume that z E y. Then y R x and 99 y R z. By symmetry, x R y, and by transitivity, x R z. Therefore, z e I. and z & X. then z ¢ y.
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