For the set A= [-0, 4) U{5,9} U[6,7] , 6 is interior point of A ylgn İhi

Q: Suppose that X,Y and Z are subsets of {1, 2, 3, . . . , 10} and |X| = |Y| = |Z| = 7. (i) Prove that…

A: Given that X,Y,Z are subsets of {1,2,3,...,10} |X|=|Y|=|Z|=7. X∪Y≤10 i)X∩Y≥4 X∪Y=X+Y-X∩YX∩Y=X+Y-X∪Y…

Q: j) A U A = A k) A U $ = A

A:

Q: Let P1, P2, P3, .. , P78 be some coplanar points with k lines on the same plane. If exactly two line…

A: Given,  P1, P2,P3,...,P78 be some coplanar points with k lines on the same plane. If exactly…

Q: Show that the subset of R2 given by {(x, y) E R² : xy 1, x > 0} is closed.

A:

Q: Given the bases B = {u,,u2} and B'={u'1, u'2} for R², where and u'1= The transition mat rix Pg-8'…

A: Given bases  are B=u1,u2 and B'=u1',u2'  where u1=10,u2=01 and u1'=12,u2'=11 To find the transition…

Q: Let u= and v= Show that is in Span (u, v} for all x and y.

A: Given u=4-1 , v=41 and b=xy. The vector b=xy is in span u,v if the system containing u,v and b is…

Q: Show by any valid method except Venn diagram that ((A U D) N B)n ((C U B) N A) = AN B. %3D

A: Given: A∪D∩B∩C∪B∩A We have to prove A∪D∩B∩C∪B∩A=A∩B

Q: Prove that the four points P(2, ;1, 1), Q(1, 3, ; 2), R((2, 1, ; 3) and S(3, 2, 0) are coplanar

A: Given four points are  P(2,1,1)Q(1,3,2)R(2,1,3)S(3,2,0)

Q: 6. Show that the sets S = (1,5) and T = (0, 00) are equinumerous by finding a bijection from S to T.…

A:

Q: 1. Show that S, has the presentation (x, y|x² = y = (xy)² = 1). %3D

A: The symmetric group S3 is given as S3=I, 1 2, 1 3, 2 3, 1 2 3 1 3 2 where  I=123123, 1 2=123213, 1…

Q: Let X, Y be two sets. Prove that X × Y C P(P(X U Y )).

A:

Q: Consider two sets, A = {1, 2, 3, ..., 100} and B = {3, 5, 7, 9, ... , 201}. Prove that |A| = |B| by…

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Q: Show that S {x€R* |2,• 24. • O} is/isn't %3D convex

A:

Q: Show that the subset of R2 given by {(x, y) E R² : xy = 1, x > 0} is closed.

A:

Q: F, there is a disconnection A, B of F such that x E A and y E B. 8.X. If C1, C2 are connected…

A: We use the fact that a set X is connected if there is no continuous function form f from X to Y=…

Q: Show that the following pairs of sets S and T are equinumerous by finding a specific bijection…

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Q: of all Z € C that z iz e = e a Descnibe and sketch the set

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Q: Prove that on average, permutations of length n have H, cycles, where Hn i=1

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Q: What is u × w, assuming that w × u = 〈2, 2, 1〉?

A: Given,w×u=2,2,1 To find:u×w Solution: We know ,by property of cross product of two vectors (say,a…

Q: Let S be the subset of the ℝ2 given by all pairs (x, y) so that x2 + y2 = 2. Show that S is a closed…

A:

Q: Let u = and v = Show that is in Span (u, v} for all x and y. y -1 5

A:

Q: find the'mters ection d Exe sy,z]= [a,0,31++ 4,-2,178 [+sy,z]=[6,-12,--11]+5 [1,2,4]?

A:

Q: Suppose (X,S,u) is a measure space and f,h: X → F are S-measurable Prove that I|fh||, < ||f||, ||4|a…

A: Given X,S,μ is a measure space and the functions f,h:X→F are S-measurable. Suppose 1p+1q=1r,…

Q: Consider the set S = {zeC:0< Re(z) < 2 & 0<Im(z) < 3} What is its interior? {zeC:0<Re(z) < 2 & 0<…

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Q: 5. Let H= [ (û ¿)I a,b,c EIRI, show that: (H,+) is a sub gromp of (M,+).

A:

Q: Find r, s, and t so that (a) z = 1 2 x, (b) z + u = x, (c) z − x = y.

A: Given  And z+u=x and z-x=y. We need to find the value of r , s and t.

Q: Consider the collection of sets {Bn : n E N}, such that Vn EN, B, = {x € R : -n <a <n}. Find U B,…

A: We are given, So, B1 = { x ∈ R | -1 ≤ x ≤ 1},       B2 = { x ∈ R | -2 ≤ x ≤ 2},       B3 = { x ∈ R…

Q: a subipace of Masa And, the set s-{(:),(;:), () #) w {():x, YaR} i) W = {Ğ ): x, Question 3 -{:9),(?…

A:

Q: Prove that on average, permutations of length n have H, cycles, where Σ Hn i=1

A: We will take an arbitrary set containing n elements and observe the required permutations.

Q: 4. Let r E R" be such that r = (E1,... , En). Show that the norms ...+ En and = V are equivalent so…

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Q: Let T={[a, ) : a€R}, then T is not Borel algebra.

A:

Q: Show whether the sets are connected or not. 1. {z € C | 1< ]z| < 2} 2. D(0, 1) U D(2i, }),

A: 1. Given: D⊂E⊂A then D=E    To Find: To whether the sets are connected.   Where z=x+iy           z…

Q: Suppose that X,Y, Z CU. If n(U)= 37, n(X) = 19, n(Y) = 11, n(Z) = 16, n(X nY) = 8, n(X n Z) = 12,…

A: NOTE: A is X, B is Y and C is Z.

Q: Let V C(R ) be a nonempty open set satisfying a – b EV whenever a, b e V. Prove that V = R. %D

A:

Q: 6. Given plane M parallel to plane K. A and Care points in M, and B and D are points in K such that…

A: As per bartleby guidelines only 1 question is answered, please upload each question separately: It…

Q: Let {C, : r € [0, 0)} be the family of indexed sets given by Cr {(x, y) E R : a? + y = r²} a)…

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Q: Find the intersection of the sets. {w, y, z}∩∅

A: Given, w,y,z∩∅we have to find intersection of the sets.

Q: Let X, Y be two sets. Prove that X x YC P(P(X UY )).

A:

Q: Find all maps ¢:(Z,+) → (Q*, ×) satisfying O (a + b) = ¢(a) × ¢(b) Va, b e Z

A: We derive the function from the given properties.

Q: Show that U. AS cuhere N= {3 unaluralnun %3D

A: To proveG={N,*} is not a group under binary operation a*b=ab where a,b∈N

Q: • Show that if A is compact and Bis closed, and d(A, B) = 0, then ANB÷Ø.

A:

Q: Let the non-empty set S = {x - yv3:x,y ER}, show that S is closed'under multiplication then find the…

A: We have to show that the given set S=x-y3:x,y∈R is closed under multiplication and also find the…

Q: Let 2 z, and Z2 = %3D ) show {2, ,2z} that or thonormal c's an set in O Write the rector as a 2 = -…

A:

Q: A subset W of R° is given by = {(x,y, z) | 2x = y = 2z}. (i) Write down the distinct elements of W.

A:

Q: Let X = {a, b, c, d, e} and let T ={X, Ø, {a}, {c,d}, {a,c,d), {b,c,d,e}), therm %3D O (X,T) is…

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Q: 10. Prove that the set H={[,ne subgroap of GL(2, R). For show that x HxcH but xHx+H. :9 GL(2,R),

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Q: Prove that S UT = SN T for all sets S and T using the rules for complements of sets.

A: rules of complements x∈A⇔x∉A¯

Q: Let A = {x E Z* : x is a mutiple of 2 and x < 15}. Rewrite A in roster notatlon.

A: Given that : A = {x∈Z+ : x is a multiple of 2 and x &lt;15 }   Roster form : A representation of a…

Q: Prove that if A C R and t > 0, then |A| = |AN(-t,t)|+|AN (R\(-t,t))|-

A:

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”.