V.1. What is the cardinality of the set {r € Z|r = y°, for some y € Z, – 10 < y < 10}
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: 5. Define a relation S from R to R as follows: For all (x,y) ERx R, (x,y) €S means that x > y. a. Is…
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Q: e following sets of points is collinear 2. L(-5,13), M(-10,10), N(-20,4)
A: Here we have to prove that each of the following sets of points is collinear. L(-5,13); M(-10,10),…
Q: 1.* Let U C R? be an open and bounded set and u e C²(U) n C(U) satisfy -1 Ди(х) : in U. 1 + |x|
A: Since I have solved similar ones, Let us show that u attains its minimum on the boundary
Q: Question 13 How many elements are there in the relation R on the set {0, 1, 2, 3, . . . , 100},…
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Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is F open?
A: A set is open if and only if it contains all of its limit points. Also recall that the union of open…
Q: 2. Let x be an element of a convex set S. Assume that x1 = x + €ip E S and x2 = x – €2P E S, where p…
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Q: 8. Sketch the sets X = {(x, y) e R :x+y s1} and Y = {(x,y) ER:-1sys0} on R. On separate drawings,…
A: And,
Q: X, Y and Z be subsets of the set . Verify the following propert XU(YUZ) = (XUY)UZ, XN(YnZ) = (XnY)…
A: X, Y, Z be subset of a universal set Ω to verify…
Q: 6. Sketch the set X = {(x, y) E R? : y<x²} on R2. Shade in the set X. %3D
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Q: Let X, Y be two sets. Prove that X × Y C P(P(X UY )).
A: It is given that X,Y be two sets. To show that X×Y⊆PPX∪Y Firstly, we define the set X×Y as, X×Y=x,y…
Q: How many 5-tuples (x, y, z, u, v) of natural numbers satisfy x + y+ z+u + v = 20? Let x= X-1,…
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Q: Suppose that |A| = m and |B| = n. Find the following cardinalities. a. b. Ip(A x B)| c. Ip(A) x…
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Q: O Show that if Si and S2 are convex sets in R™X", then so is t mxn S = {(x, y1 + y2) | x € R", y1,…
A: Convex set: , If a and b are points in a vector space S. Then the vector space is called convex set…
Q: Determine whether is a linearly independent set of vec- 3 11 tors in R3.
A: A set of vectors v1,v2,v3,…,vk in a vector space is said to be linearly independent if…
Q: Let X, Y be two sets. Prove that X × YC P(P(X U Y )).
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Q: Let X and Y be two discrete spaces, then * X is homomorphic to Y if and only if X and Y are both…
A: Two discrete spaces are homeomorphic if there exists a bijection between them and it is possible if…
Q: We defined the relation between sets by A ~ B means that there there is a 1-1 correspondence ƒ : A →…
A: We define the relation ~ between sets by A~B means that there is a one-one correspondence f:A→B Show…
Q: Suppose that |A| = m and |B| = n. Find the following cardinalities. a. P(P(P(A))). b. P(A × B)| с.…
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Q: Problem 12.2.8. Suppose X is an uncountable set and Y C X is countably infinite. Prove that X and X…
A: Given that X is a uncountable set and Y⊂X is a countably infinite set. To find : X and X-Y has same…
Q: Prove that if ø is convex, then a-[$(x + Az) – Þ(x)] is nondecreasing for å > 0.
A: As φ is convex we conclude that its second derivative : φ'' is always positive ∴ φ''(x) > 0…
Q: of points of S. 6.30 Show that a set consisting of a finite number of points in E is bounded. Gnita…
A: Given, a set consisting of a finite number of points in E.
Q: The relation - on the set A = Z x (N - {0}) by (u, v) - (z, w) if and only if wu = zv. Specify the…
A: We will use the relation wu=zv between two elements (u,v) and (z,w) .
Q: For the following sets, determine its cardinality. (ℝ^2)\(ℚ×ℤ)（ is Cartesian product）
A: We find cardinality. (ℝ^2)\(ℚ×ℤ)（ is Cartesian product）
Q: A set SCR" is bounded if there exists k €R such that, for all x E S, || = VE, 0. What can you say…
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Q: C.1 Let r > 0 and define the closed ball by B,(x) = {y € X : d(x,y) <r} Show that B(x) is closed set…
A: Given: τ=B⊆X : ∀x0∈B, ∃rx0>0, such that Brx0x0⊂B is a topology in X. Open ball Br(x)=y∈X : d(x,…
Q: Related to the solution for exercise problem 12, in section 7.4 of textbook "Discrete Mathematics:…
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Q: Let X and Y be two discrete spaces, then * X is never homeomorphic to Y X is homomorphic to Y if and…
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Q: 4.4. Let S be a nonempty set. Suppose that to each ordered pair (x, y) E S x S a nonnegative real…
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Q: IF f:x-у and vCy proof fy/v)=x/f"(v) and proof f is continous iff f'(E) is closed set in x for every…
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Q: 3. ** Let X, Y be finite sets, and f is a function XY. What can you say about the cardinality if X,…
A: 3. Given X, Y are finite sets and f:X→Y is a function To Comment about cardinality of X and Y if…
Q: Given two sets X and Y, we denote the cardinality of X by |X| and the set of all functions from X to…
A: We are given two sets X and Y. The set of all functions from X to Y is denoted by YX. Suppose that…
Q: 2.6.4 Let A CR be a non – empty set. Suppose that A is bounded above. Let U = {x E R|x is an upper…
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Q: Let X and Y be two discrete spaces, then * X is homomorphic to Y if and only if X and Y are both…
A: Our guidelines we are supposed to answer only one question. Kindly repost other question as the next…
Q: Let S be the set of integers. If a, b E S, define aRb if ab >= 0. Is R anequivalence relation on…
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Q: Prove that if R, S, and T are nonempty sets, then R×T ⊆ S×T if and only if R ⊆ S.
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Q: 2 49) Given nonempty subsets of R², say Y....., let Y = nin enner-xer Fix P E R². For a nonempty…
A: Given: Non empty subset To find: Solution
Q: 1.39. For each point (m, n) in the digital plane, determine the smallest closed set containing (m,…
A: given: for an each point (m, n) in the digital planeTo…
Q: Identify all nonempty set/s from the following A = {x|x € N, 2x + 7 = 3} B = {x|x € Z, 2x + 7 = 4} C…
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Q: [2.13] Let S be a nonempty set in R". Show that S is convex if and only if for each integer k > 2,…
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Q: Let X, Y be two sets. Prove that X x YC P(P(X UY )).
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Q: 7. Suppose that a set X is equipped with the finite closed topology. Which one of the followings is…
A: As per the policy, we are solving first question. Please repost the question and specify which…
Q: Let X and Y be two discrete spaces, then * X is homeomorphic to Y if and only if X and Y have the…
A: X and Y are two discrete spaces then X and Y The topology is discrete then for x∈X is an open…
Q: of E. 4. Let E CR be a bounded set. Prove sup E e Ë.
A: (4) Let E be a bounded subset of R and let u be the sup E. Then two cases arise- Either u belongs…
Q: e. If x is a subset of V of cardinality 2, then f (x) is the edge connecting the two vertices in x.
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Q: Match the following sets with their cardinalities. {t, 5, 1, p} Choose... {0} Choose... Choose... {}…
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Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is F closed?
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Q: 4. Prove that the sets (–1,5) and [-1,1) U {2} U (3, ∞) have the same cardinality
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Q: We defined the relation between sets by A - B means that there there is a 1-1 correspondence f : A →…
A: The complete solutions is in given below
Q: 4.2 Let Fj, be a closed set for k = 1,2, ... ,n in (S, d). Show that U Fk is closed.
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