1. (a) For a set A, P (A) is the set of subsets of A.Write down all the different elements of the set P (P (0)).(b) Proof. for all sets A and B holds B\ (B \A) = A if and only ifACB.(c) Proof: the set Z and the set {2:|i€Zs } are equal.2. (a) Investigate with a truth table whether (P = (Q VR)) = ((PA-Q) = R) ais tautology.(b) Proof or refutation: - (V xeN 3 yez ((x> y)^ (x 2 =y 2 ).(c) Given is the operation D on Z defined by D: (a, b) → - a+b- 1. Determine theneutral element of this operation. Proof or rebuttal: D is associative.(d) Given is the relationship {(x, y) € (N >0 × N >0 ): |x - y| 0 x N >0. Wenote this relationship with ~. Proof: v is an equivalence relationship.

Note: We are entitled to solve only the first question, unless specified 1) (a) We know the power…

1. (a) For a set A, P (A) is the set of subsets of A. Write down all the different elements of the set P (P (0)). (b) Proof: for all sets A and B holds B\ (B \A) = A if and only if ACB. (c) Proof: the set Z and the set {2 :|iE Z s } are equal.

1. (a) For a set A, P (A) is the set of subsets of A. Write down all the different elements of the set P (P (0)). (b) Proof: for all sets A and B holds B\ (B \A) = A if and only if ACB. (c) Proof: the set Z and the set {2 :|iE Z s } are equal.

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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Concept explainers

Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.

Question

1. (a) For a set A, P (A) is the set of subsets of A.
Write down all the different elements of the set P (P (0)).
(b) Proof. for all sets A and B holds B\ (B \A) = A if and only if
ACB.
(c) Proof: the set Z and the set {2:|i€Zs } are equal.
2. (a) Investigate with a truth table whether (P = (Q VR)) = ((PA-Q) = R) a
is tautology.
(b) Proof or refutation: - (V xeN 3 yez ((x> y)^ (x 2 =y 2 ).
(c) Given is the operation D on Z defined by D: (a, b) → - a+b- 1. Determine the
neutral element of this operation. Proof or rebuttal: D is associative.
(d) Given is the relationship {(x, y) € (N >0 × N >0 ): |x - y| <x y} at N >0 x N >0. We
note this relationship with ~. Proof: v is an equivalence relationship.

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