4. Proving by Example. Let U = {x|0 < x < 10}, A = { x €U|x is odd), and B:= {re U x is even).(a) Using A and B show that De Morgan's Law is true.(b) Using A and B Show that the remark number 7 on the lecture on inter-sections of Set is true.

This is a problem of set theory.

Answered: 4. Proving by Example. Let U := { x | 0…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Consider the set U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and define subsets
None
6. The following statements about sets are false. For each statement, give an example, i.e., a choice of sets, for which the statement is false. Such examples are called counterexamples. They are examples that are counter to, i.e., contrary to, the assertion. (a) AUBCAn B for all A, B. (b) ANØ: = A for all A. (c) AN (BUC) = (A ^ B) UC for all A, B, C.
Choose the correct option. Let S be the set of points whose abscissas and ordinates are natural numbers . Let P∈S such that the sum of the distance of P from (8,0) and (0,12) is minimum among all elements in S . Then the number of such points P in S is- (a)1 (b)3 (c)5 (d)11.
1. Find the union C1 U C2 and the intersection C1n C2 of the two sets C1 and C2, where (a) C1 = {0, 1, 2, }, C2 = {2,3, 4}. (b) C1= {x:0 < x <2}, C2= {x :1
DISCRETE MATHEMATICS
1. Let A = {-30, -10, 10, 30, 50} and B = {-20, -10, 0, 10} be subsets of the universal set ε = {10x x € ZZ and -3≤ x <6}. (a) List the elements of An B and (AUB)'. (b) Evaluate: (i) │A\ B|; (ii) |A × B|; [1] (iii) |P(A)|; (iv) P(B) × P(A)|. [2] (c) Are the following statements are true or false? Explain your reasoning. (i) 0 СА; (ii) {0} = P(A); (iii) {10} E AUB; (iv) {10} is a proper subset of An B. (d) List the elements of P(A) P(B). [2] [2] (e) If the set C is also a subset of Ɛ, with |C| = 5, An C = {-10, 30} and BNC {-10, 20}, list the elements of C. = [2]
4
Proposition 3.4 Let E be a measurable subset of R. (i) ƒ : E → R is measurable if and only if both f+ and f- are measurable. (ii) If ƒ is measurable, then so is |f|; but the converse is false. Hint Part (ii) requires the existence of non-measurable sets (as proved in the Appendix) not their particular form.

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4. Proving by Example. Let U := { x | 0 < x < 10}, A = { 1 € U|x is odd), and B:= {xe U|x is even}. (a) Using A and B show that De Morgan's Law is true. (b) Using A and B Show that the remark number 7 on the lecture on inter- sections of Set is true.