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2. Let U be a set, and let A CU. Recall the indicator function XA: U...
Let U be a set, and let A CU. Recall the indicator function XA: U +...
Let U be a set, and let A CU. Recall the indicator function XA: U + Z, defined by XA(x) = ſi, rEA 0, A. Now, let A, B CU and consider the symmetric difference of A and B defined by A AB= (A - B)U(B - A). (a) Show that AAB CU, and compute Ø A A. (b) Prove that Ve EU, XAAB(C) = XA(2) + XB(2), where addition is taken modulo 2 (so that 1+1 = 0).
5. Let A, B, C be subsets of a universal set U. Recall for D CU...
5. Let A, B, C be subsets of a universal set U. Recall for D CU that XD denotes the characteristic function of D. Prove that XAUBUC = XA +XB+XC - XAMB - XAOC · XBNC + XANBNC. Hint: Facts that you may use: (1) Xpe = 1 – Xd. (2) (AU BUC)° = A n Bºn Cº. (3) XEnF = XEXF. (4) XEnFnG XEXFXG. Don't prove these facts.
Let A, B, C be subsets of a universal set U. Recall for D C U...
Let A, B, C be subsets of a universal set U. Recall for D C U that XD denotes the characteristic function of D. Prove that XAUBUC = XA + XB+XC - XACB - XAC - XB C +XAOBOC. Hint: Facts that you may use: (1) XD 1-XD. (2) (AU BUC)° = ACB 1C. (3) XEnF = XEXF. (4) XEnFnG = XEXFXG. Don't prove these facts.
Let X be a set and let T be the family of subsets U of X such that X\U (the complement of U) is at most countable, together with the empty set. a) Prove that T is a topology for X. b) Describe the con...
Let X be a set and let T be the family of subsets U of X such
that X\U (the complement of U) is at most countable, together with
the empty set. a) Prove that T is a topology for X. b) Describe the
convergent sequences in X with respect to this topology. Prove that
if X is uncountable, then there is a subset S of X whose closure
contains points that are not limits of the sequences in S....
Definition 10.2.3 If A is a set, then the indicator function of A, denoted by I, is defined as 1 ...
Definition 10.2.3 If A is a set, then the indicator function of A, denoted by I, is defined as 1 if x e A 5. , X, be a random sample from a gamma distribution, X, ~GAM(0, 2),Show Let X1, that x, is sufficient for θ (a) by using equation (10.2.1), (b) by the factorization criterion of equation (10.2.3) oren. 0.2.1 Factorization Criterion If X1, , X" have joint pdff(x,, , x.. θ), and is (S S then S..., S...
Please do problem 14.12 Problem 14.6. Let X be a nonempty set and let A be a subset of X. The character- istic function...
Please do problem 14.12
Problem 14.6. Let X be a nonempty set and let A be a subset of X. The character- istic function or indicator function of the set A in X is 1 if x E A XA(x)-10 if xeX\A A-X→ {0,1} defined by Problem 14.12. See Problems 14.6, 14.7, and 14.11 for the definitions (a) Write the greatest integer function as a sum of characteristic functions (there may be more than one way to do this). Depending...
1. Let A= {0,1}2 U... U{0,1}5 and let < be the order on A defined by...
1. Let A= {0,1}2 U... U{0,1}5 and let < be the order on A defined by (s, t) E< if and only if s is a prefix of t. (We consider a word to be a prefix of itself.) (a) Find all minimal elements in A. (Recall that an element x is minimal if there does not exist y E A with y < x.) (b) Are 010 and 01101 comparable? 2. Give an example of a total order on...
probelms 9.1 9 Modular arithmetic Definition 9.1 Let S be a set. A relation R =...
probelms 9.1
9 Modular arithmetic Definition 9.1 Let S be a set. A relation R = R(,y) on S is a statement about pairs (x,y) of elements of S. For r,y ES, I is related to y notation: Ry) if R(x,y) is true. A relation Ris: Reflexive if for any I ES, R. Symmetric if for any ry ES, Ry implies y Rr. Transitive if for any r.y.ES, Ry and yRimply R. An equivalence relation is a reflexive, symmetric and...
Problem 2: (Topological Characterization of Continuity) Let : R → R be a function. Recall that...
Problem 2: (Topological Characterization of Continuity) Let : R → R be a function. Recall that for a subset BCR, we have the set (B) := ER: (a) e B). Prove that is continuous if and only if f'(U) is open for all open sets U CR. Hint: you can use the characterizations of continuity from Theorem 4.3.2 in our textbook
Let x=(3,1,0], and let U be the subspace spanned by the orthogonal set {[1, -2, 1],[1,...
Let x=(3,1,0], and let U be the subspace spanned by the orthogonal set {[1, -2, 1],[1, 1, 1]}. Vector x can be written as X=X1+X2, where x1 is in U and xz is orthogonal to U. Find the first coordinate of x2.
Let U be a set, and let A CU. Recall the indicator function XA: U + Z, defined by XA(x) = ſi, rEA 0, A. Now, let A, B CU and consider the symmetric difference of A and B defined by A AB= (A - B)U(B - A). (a) Show that AAB CU, and compute Ø A A. (b) Prove that Ve EU, XAAB(C) = XA(2) + XB(2), where addition is taken modulo 2 (so that 1+1 = 0).
5. Let A, B, C be subsets of a universal set U. Recall for D CU that XD denotes the characteristic function of D. Prove that XAUBUC = XA +XB+XC - XAMB - XAOC · XBNC + XANBNC. Hint: Facts that you may use: (1) Xpe = 1 – Xd. (2) (AU BUC)° = A n Bºn Cº. (3) XEnF = XEXF. (4) XEnFnG XEXFXG. Don't prove these facts.
Let A, B, C be subsets of a universal set U. Recall for D C U that XD denotes the characteristic function of D. Prove that XAUBUC = XA + XB+XC - XACB - XAC - XB C +XAOBOC. Hint: Facts that you may use: (1) XD 1-XD. (2) (AU BUC)° = ACB 1C. (3) XEnF = XEXF. (4) XEnFnG = XEXFXG. Don't prove these facts.
Let X be a set and let T be the family of subsets U of X such
that X\U (the complement of U) is at most countable, together with
the empty set. a) Prove that T is a topology for X. b) Describe the
convergent sequences in X with respect to this topology. Prove that
if X is uncountable, then there is a subset S of X whose closure
contains points that are not limits of the sequences in S....
Definition 10.2.3 If A is a set, then the indicator function of A, denoted by I, is defined as 1 if x e A 5. , X, be a random sample from a gamma distribution, X, ~GAM(0, 2),Show Let X1, that x, is sufficient for θ (a) by using equation (10.2.1), (b) by the factorization criterion of equation (10.2.3) oren. 0.2.1 Factorization Criterion If X1, , X" have joint pdff(x,, , x.. θ), and is (S S then S..., S...
Please do problem 14.12
Problem 14.6. Let X be a nonempty set and let A be a subset of X. The character- istic function or indicator function of the set A in X is 1 if x E A XA(x)-10 if xeX\A A-X→ {0,1} defined by Problem 14.12. See Problems 14.6, 14.7, and 14.11 for the definitions (a) Write the greatest integer function as a sum of characteristic functions (there may be more than one way to do this). Depending...
1. Let A= {0,1}2 U... U{0,1}5 and let < be the order on A defined by (s, t) E< if and only if s is a prefix of t. (We consider a word to be a prefix of itself.) (a) Find all minimal elements in A. (Recall that an element x is minimal if there does not exist y E A with y < x.) (b) Are 010 and 01101 comparable? 2. Give an example of a total order on...
probelms 9.1
9 Modular arithmetic Definition 9.1 Let S be a set. A relation R = R(,y) on S is a statement about pairs (x,y) of elements of S. For r,y ES, I is related to y notation: Ry) if R(x,y) is true. A relation Ris: Reflexive if for any I ES, R. Symmetric if for any ry ES, Ry implies y Rr. Transitive if for any r.y.ES, Ry and yRimply R. An equivalence relation is a reflexive, symmetric and...
Problem 2: (Topological Characterization of Continuity) Let : R → R be a function. Recall that for a subset BCR, we have the set (B) := ER: (a) e B). Prove that is continuous if and only if f'(U) is open for all open sets U CR. Hint: you can use the characterizations of continuity from Theorem 4.3.2 in our textbook
Let x=(3,1,0], and let U be the subspace spanned by the orthogonal set {[1, -2, 1],[1, 1, 1]}. Vector x can be written as X=X1+X2, where x1 is in U and xz is orthogonal to U. Find the first coordinate of x2.
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