Shown again is the table indicating the marital status of the U.S. population in 2015. Numbers in the table are expressed in millions. Use the data in the table to solve Exercises 65-76. Express probabilities as simplified fractions and as decimals rounded to the nearest hundredth. MARITAL STATUS OF THE U.S. POPULATION, AGES 15 OR OLDER, 2015, IN MILLIONS Married Never Married Divorced Widowed Total Male 66 43 11 3 123 Female 67 38 15 11 131 Total 133 81 26 14 254 If one person is selected from the population described in the table, find the Probability, that the person 69. is male or divorced. 69 127 = 0.54
Shown again is the table indicating the marital status of the U.S. population in 2015. Numbers in the table are expressed in millions. Use the data in the table to solve Exercises 65-76. Express probabilities as simplified fractions and as decimals rounded to the nearest hundredth. MARITAL STATUS OF THE U.S. POPULATION, AGES 15 OR OLDER, 2015, IN MILLIONS Married Never Married Divorced Widowed Total Male 66 43 11 3 123 Female 67 38 15 11 131 Total 133 81 26 14 254 If one person is selected from the population described in the table, find the Probability, that the person 69. is male or divorced. 69 127 = 0.54
Shown again is the table indicating the marital status of the U.S. population in 2015. Numbers in the table are expressed in millions. Use the data in the table to solve Exercises 65-76. Express probabilities as simplified fractions and as decimals rounded to the nearest hundredth.
MARITAL STATUS OF THE U.S. POPULATION, AGES 15 OR OLDER, 2015, IN MILLIONS
Married
Never Married
Divorced
Widowed
Total
Male
66
43
11
3
123
Female
67
38
15
11
131
Total
133
81
26
14
254
If one person is selected from the population described in the table, find the Probability, that the person
(1) Let F be a field, show that the vector space F,NEZ* be a finite dimension.
(2) Let P2(x) be the vector space of polynomial of degree equal or less than two
and M={a+bx+cx²/a,b,cЄ R,a+b=c),show that whether Mis hyperspace or not.
(3) Let A and B be a subset of a vector space such that ACB, show that whether:
(a) if A is convex then B is convex or not. (b) if B is convex then A is convex or not.
(4) Let R be a field of real numbers and X=R, X is a vector space over R show that by
definition the norms/II.II, and II.112 on X are equivalent where
Ilxll₁ = max(lx,l, i=1,2,...,n) and llxll₂=(x²).
oper
(5) Let Ⓡ be a field of real numbers, Ⓡis a normed space under usual operations and
norm, let E=(2,5,8), find int(E), b(E) and D(E).
(6) Write the definition of bounded linear function between two normed spaces and
write with prove the relation between continuous and bounded linear function
between two normed spaces.
ind
→ 6
Q₁/(a) Let R be a field of real numbers and X-P(x)=(a+bx+cx²+dx/ a,b,c,dER},X is
a vector space over R, show that is finite dimension.
(b) Let be a bijective linear function from a finite dimension vector ✓ into
a space Yand Sbe a basis for X, show that whether f(S) basis for or not.
(c) Let be a vector space over a field F and A,B)affine subsets of X,show that
whether aAn BB, aAU BB be affine subsets of X or not, a,ẞ EF.
(12
Jal (answer only two) (6) Let M be a non-empty subset of a vector space X and tEX,
show that M is a hyperspace of X iff t+M is a hyperplane of X and tЄt+M.
(b) State Jahn-Banach theorem and write with prove an application of Hahn-
(b) Let A and B be two subset of a linear space X such that ACB, show that
whether if A is affine set then B affine or need not and if B affine set then A affine set
or need not.
Qz/antonly be
a-Show that every hyperspace of a vecor space X is hyperplane but the convers
need not to be true.
b- Let M be a finite dimension subspace of a Banach space X show that M is closed set.
c-Show that every two norms on finite dimension vector space are equivant (1)
Q/answer only two
a-Write the definition of bounded set in: a normed space and write with prove an
equivalent statement to a definition.
b- Let f be a function from a normed space X into a normed space Y, show that f
continuous iff f is bounded.
c-Show that every finite dimension normed space is a Banach.
Q/a- Let A and B two open sets in a normed space X, show that by definition
AnB and AUB are open sets.
(1
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