4. (10 points) Show that if A is an infinite set, then whenever B is a set, AUB is also an infinite set.

Q: Define a smallest possible set of which both {1,3,5} and {0, 3,4 } are subsets.

A: The subsets given are {1, 3, 5} and {0, 3, 4}.

Q: Thoorem: is not closed set

A:

Q: Give an example of a set in R ^ 2 with the usual topology that is not connected but has a connected…

A: Let's examine the  problem for path-connectedness example of a set in R2 with the usual topology…

Q: Suppose f function between infinite sets A and B, f is not onto function, does the sets A and B have…

A: Yes. It is possible that A and B have same cardinality. Let A and B both are the set of natural…

Q: Give an example of each of the following or explain why you think such a set could not exist. (i) A…

A: (i). Such a set could not exist. Let  be a nonempty subset of a topological space . Let . If  is not…

Q: define a set of which both {1,3,4} and {0,3,5} are subsets find such a set with the smallest…

A: We have to find the set with smallest possible number of elements which both {1,3,4} and {0,3,5} are…

Q: Prove or disprove: For all sets A and B, An(AUB)

A: We have to prove or disprove , for all sets A and B A∩(A∪B)=A.

Q: If AnB is pathwise connected set.  then A and B are:  a. Pathwise connected sets b. Not pathwise…

A: Pathwise connected means any two points in that set can be join by a continuous function. Roughly…

Q: 8. Disprove that every compact set has only a finite number of isolated points. V Note that the…

A: Let C be the compact set.If possible suppose that 'C' has a infinitely many isolated points. say…

Q: 3.1 Show that a finite intersection of closed sets is closed.

A:

Q: 26

A: The objective is to find an infinite collection of closed sets whose union is not closed.

Q: Show

A: We prove this statement by using definition of Borel Set.

Q: Write down the non-empty proper subset of the set {a,b,c}. Make sure to use correct notation…

A:

Q: 5. Give an example (by words) for a collection of objects which is not well defined

A: Given, We have to write an example for a collection of objects which is not well defined.

Q: Prove that (2,5) is an open set

A:

Q: 4) Using equality of sets, prove that (An B)' = A' U B'

A: we know thatA∪B=x: x∈A or x∈BA∩B=x: x∈A or x∈BA'=x:x∈U, x∉A=U \ AWhere U is a universal set.That is…

Q: Give an example of a closed set in R ^ 3 with the usual topology that is not compact

A: Given the usual topology on ℝ3. i.e. τusual=U⊆ℝ3: ∀x∈U, ∃∈>0 such that B∈x⊆U

Q: By considering the collection of intervals {(-) : n € N} show that the intersection of an infinite…

A: Let  Sn=-1n,1n :n∈N      where N is set of natural number. for n=1,2,3,...... S1=-1,1…

Q: Give an example of a bounded set that has infinitely many isolated points

A:

Q: (a) What is the cardinality of the interval (0, 1), finite, countably infinite, or uncountable? (b)…

A:

Q: subset

A:

Q: List all the possible subsets of set G, where G = {-7,m,z}- %3D

A:

Q: 1. Give at least 5 pairs of numbers a and b. For each of those pairs find at least 10 elements of…

A: We need to find at least 5 pairs of numbers a and b. For each of those pairs  at least 10 elements…

Q: how that if |A| = |B| and |B| = |C| then |A| = |C|. The sets A, B, and C may be infinite, so ou must…

A:

Q: 15. We have seen that an infinite set can be put into a one-to-one correspondence with a proper…

A:

Q: Prove that a nonempty set is bounded if and only if it is contained in a bounded interval

A: A non empty set S is called bounded if it has both upper and lower bounds, In a certain sense a non…

Q: 5. Show that a finite union of countable sets is countable.

A: The answer is given as follows :

Q: Suppose we modify the construction of the Cantor set by starting with the open interval (0, 1) and…

A:

Q: Show that any subset of bounded set is bounded

A:

Q: Write down the non-empty proper subset of the set {a,b,c}. Make sure to use correct notation…

A: Consider the given set A = {a, b, c}. A proper subset of a set A is a subset of A that is not equal…

Q: 7. Show that the Obenchain class LOb(B) is a convex set.

A: Given: To show the Obenchain class LOb (β) is a convex set is given as,

Q: 6. Give an example of a set S which is (i) bounded above but not bounded below, (ii) bounded below…

A: (i). Let  S = { ..., -5, - 4, - 3, - 2, - 1 }

Q: Prove that the smallest member of a set, if exits, is the greatest lower bound of the set. Correct…

A:

Q: ind the all possible number of different multi sets of size '4' from -element set, in which at least…

A: In this question, the concept of permutation is applied. Permutation Permutations are frequently…

Q: The union of a infinite number of closed set need not be a Closed set Open Set Both open and close…

A: The union of infinite number of closed sets is

Q: Find the cardinality of the set {x|x is an even whole number less than 10}.

A:

Q: Explain the idea of cardinality. When is a set denumerable? Show that the union of two denumerable…

A:

Q: ) Find an example of an infinite collection of closed sets whose union is not closed.

A: Any closed interval is closed in R. From that we created this example.

Q: 5. State whether each of the following statements is true or false, in either case substan- tiate/…

A:

Q: Let A and B be infinite sets that are countable. Do A and B necessarily have the same cardinality as…

A: Given question is Let A and B be infinite sets that are countable. Do A and B necessarily have the…

Q: 5. (a) How many functions are there from a set with n elements to a set with m elements?

A:

Q: The intersetion of in finite open sets heed not to be open.

A:

Q: 2. Show that the set N has no limit point.

A:

Q: Any two closed intervals has the different cardinality

A: we know that any two closed intervals has the same cardinality

Q: Give an example of a set in R ^ 2 with the usual topology that does not have any subset (not…

A: Given that R2 with the usual topology. The aim is to find an example of the set  R2 with the usual…

Q: Give example for following 1/ set is not compact

A: Compact set:- A set S⊆R is called compact “if every sequence in S has a subsequence that converges…

Q: The union of the universal set and any of its subset is * none of the choices cannot be determined…

A: our objective is to conclude