1. Let S = (0,7) U {7 +1³,... Define < on 5 by a

Q: 8. Consider the binary relation R defined on (N* x N*) by: V (x1. Y1). (X2. Y2) E N* x N*, (x1. Yı)…

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Q: HW6 Proue →1<a = | <aca

A: We have to prove that  For all a &gt; 1, we always have 1 &lt; a &lt; a2 .

Q: How many elements are in the Cartesian product U x V? * Let U = {u is odd | 9 < u < 14} and V = {v €…

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Q: 1. Let S = {1,2, 3, 4}. (a) How many relations are there on S? (b) How many are reflexive? (o) How m…

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Q: (ii) for every singleton set A, m* (A) = 0.

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Q: 5. Let R and S be relations on a set A. Use proof by contradiction to show that if R and S are…

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Q: Consider the partial order with domain{3,5,6,7,10,14,20,30,60}and with x ≤ y if x evenly divides y.…

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Q: EX: Show that the telation ? >' is PartiaL oldering set of Por R-

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Q: 4. Let W be the set of all words in the English language and let L be the set of letters {A, B,…

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Q: 6. A relation R on a set X is called a total order if it is antisymmetric, reflexive, transi- tive,…

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Q: Exercise 1. Let 0 < & ≤ 1, and A := [10] E (i) Calculate the condition number of A in the 1-, 2- and…

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Q: 11. If |A| = 3, how many relations are there on A?

A: Given A=3

Q: Let S={a,b,c,d,e,f,g} be ordered as in the following Fig. f, e C a b And let X={c,d,e} The upper…

A: Suppose B is a subset of partially ordered set A. An element m∈A is said to be upper bound of B if…

Q: 4) The relationk onthe Selof real numbers Risa partialorder relation.

A: relation R on a set A is called a partial order relation if it satisfies the following three…

Q: (4) Let T be the partial order relation on P({1,2,..., 20}) defined by ATB if and only if |A < |B|…

A: Question: Let T be the partial order relation on P1,2,...,20 defined by ATB if and only if A&lt;B or…

Q: Let < be a partial order on A. Assume that a,b EA are elements with the property that: for any X EA,…

A: (First part): Reflexive property : Since ≤ is an reflexive relation then for every element x∈A we…

Q: 6. Let 2 be a set and f, g: → R. Assume max{f(x) + g(x) | x = }, max{f(x) |x€N}, and max{g(x) | x =…

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Q: Let D be the set of all finite subsets of positive integers. Define a function T:Z* → D as follows:…

A: Since T(n)=The set of positive divisors of n. Since 17 ,5 are prime numbers.

Q: Let A = {2,6}. Consider the set B = A x P(A) (where P(A) is the power set of A). Define a partial…

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Q: 8) a) List the elements of each of the following sets, using the .' notation where necessary: i)…

A: Since it is multiple questions with multipart. As per our company guidelines, we have to solve the…

Q: draw the graph of this relation. Search this relation

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Q: (1.2.7)List all elements in the following sets. (a) T = {m € Z | m = 1 + (−1)¹ for some integer i}…

A: The sets are: (a) T=m∈Z | m=1+-1i for some integer i (b) U=r∈Z | 2≤r≤-2 (c) U=r∈Z | -2≤r≤2 To Do:…

Q: Prove that if x is a real number such that x ≥ 0 and x ≤ € for all € E R where € > 0, then x = = 0.

A: This question is from real analysis.Answer given below.

Q: 4. Define a relation R on N by aRb if a and b have no common factors except 1. (a) Give an example…

A: We have define  a relation R on natural numbers (N) by aRb if a and b have no common  factors except…

Q: Prove that the set B = {r € Q: 1² < 20 or r < 0o} is a cut.

A: In this question, we have to prove that the set,  B = x∈ℚ : x2 &lt; 2 or x &lt; 0 is a cut.

Q: 13. (9 points) Let D be the set of finite subsets of positive integers. Let S be the set of all…

A: The objective of this question is to find the set of all even factors of given numbers using the…

Q: Let S-(0(3).(3.61(3.61).(3.6.9 12) Then the poset (S) is None of these not a total order set because…

A: Total order Set -:  A Poset (A,~) is called total order if any two elements are comparable, i,e. if…

Q: 1- show that [Am, A¹] = 0 for all m and n discuss the physical meanings of this relation. 2-…

A: As per the question we are given the following commutation operations : [Am , An] [∂/∂x , x] And…

Q: 3. Let A be a finite set. In this exertise functions f, g will be of type g: A →A and f: A → A. (a)…

A: Injective and surjective function

Q: (2.1) Let b> 0 and X [-b, b). Here we admit %3D B"open" in r if and only if (0} B or (-b, b) C B.…

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Q: 6.3. Let (A, <A) and (B,<B) be two partially ordered sets. One can define on the cartesian product A…

A: Given: A, ⩽A and B, ⩽B are partially ordered sets.  A Cartesian product A x B is defined with…

Q: 3.7 Consider a fuzzy relation R CAXA where A = {a, b, c}. R a b e 1 1 1 1 1 1 d 1 1 1 a) Determine…

A: Solution :-

Q: 6. Let R be the partial order which has the Hasse diagram shown. List the elements of R.

A: R is a partial order relation means, R is reflexive, anti-symmetric &amp; transitive. If R is a…