Exercise 1.5. Prove that if A and B are sets satisfying the property that A \ B = B \ A, then it must be the case that A = B.
Exercise 1.6. Using definition (1.2.5) of the symmetric difference, prove that, for any sets A and B, A4B = (A ∪ B) \ (A ∩ B).
Exercise 1.7. Verify the second assertion of Theorem 1.3.4, that for any collection of sets {Ai}i∈I, \ i∈I Ai !c = [ i∈I A c i .
Exercise 1.5. Prove that if A and B are sets satisfying the property that A \...
I really need someone to solve and explain the last two questions. Thank you! Exercise 1.5. Prove that if A and B are sets satisfying the property that then it must be the case that A - B. Exercise 1.6. Using definition (1.2.5) of the symmetric difference, prove that, for any sets A and B, AAB - (AUB)I(AnB). Exercise 1.7. Verify the second assertion of Theorem 1.3.4, that for any collection of sets {Asher Ai iET iET Exercise 1.8. Prove...
Exercise 1.6. Using definition (1.2.5) of the symmetric difference, prove that, for any sets A and B, A (delta sign) B = (A ∪ B) \ (A ∩ B).
Exercise 1.8. Prove that, for any sets A and B, the set A ∪ B can be written as a disjoint union in the form A ∪ B = (A \ (A ∩ B)) ∪˙ (B \ (A ∩ B)) ∪˙ (A ∩ B). Exercise 1.9. Prove that, for any two finite sets A and B, |A ∪ B| = |A| + |B| − |A ∩ B|. This is a special case of the inclusion-exclusion principle. Exercise 1.10. Prove for...
Exercise 1.9. Prove that, for any two finite sets A and B, |A ∪ B| = |A| + |B| − |A ∩ B|. This is a special case of the inclusion-exclusion principle.
Theorem 16.1. Let p be a prime number. Suppose r is a Gaussian integer satisfying N(r) = p. Then r is irreducible in Z[i]. In particular, if a and b are integers such that a² +62 = p, then the Gaussian integers Ea – bi and £b£ai are irreducible. Exercise 16.1. Prove Theorem 16.1. (Hint: For the first part, suppose st is a factorization of r. You must show that this factorization is trivial. Apply the norm to obtain p=...
1.) Prove the following theorem Theorem 3.4.6. A set E C R is connected if and only if, for all nonempty disjoint sets A and B satisfying E AU B, there always erists a convergent sequence (xn) → x with (en) contained in one of A or B, and x an element of the other. (2) (10 points) Are the following claims true or false? You must use the ε-δ definition to justify your answers. x-+4 r2 16 (Here [[x]-greatest...
prove e EOLU Exercise 4.1.1. Prove Theorem 4.1.6. (Hints: for (a) and (b), use the root test (Theorem 7.5.1). For (c), use the Weierstrass M-test (Theorem 3.5.7). For (d). use Theorem 3.7.1. For (e), use Corollary 3.6.2. The signale UI tre rauUS UI CUNvergence is the IUIUWII. Theorem 4.1.6. Let - Cn(x-a)" be a formal power series, and let R be its radius of convergence. (e) (Integration of power series) For any closed interval [y, z] con- tained in (a...
1. [15 pts] Use Definition 1.5 (definition of probability function) to prove Propo- sition 1.3 () 15 pts) & (iv) [10 pts). You do not need to prove (i) and (ii). [Definition 1.5/ Let Ω be a set of all possible events. A probability function P : Ω → 0,11 satisfies the follouing three conditions (i) 0s P(A) S 1 for any event A; (iii) For any sequence of mutually exclusive events A1, A2 ,A", i.e. A, n Aj =...
Let A = {1, 2, 3} and B = {2, 3, 4, 5}. Find the cardinalities of the following sets: (i) A ∪ B (ii) A ∩ B (iii) A \ B (iv) B \ A (v) P(A ∪ B) Exercise 1.2. Let A = {◦, {◦}, {∅}} and let B = {∅, {◦}}. Find the cardinalities of the following sets: (i) A ∪ B (ii) A ∩ B (iii) A \ B (iv) A × B (v) P(A) Exercise...
1) Webber Chap. 11 Exercise 1 Prove that {a"b"c"} is not regular. Hint: Copy the proof of Theorem 11.1-only minor alterations are needed. Theorem 11.1 The language {a"b"} is not regular. • Let M = (Q, {a,b}, 8, 9., F) be any DFA over the alphabet {a,b}; we'll show that L(M) + {a"b"} • Given as for input, M visits a sequence of states: - *(q,,ɛ), then 8*(q,,a), then 8*(9,,aa), and so on • Since Q is finite, M eventually...