Let X and Y be sets. Let us say that a subset S of the cartesian product X x Y is a rectangle if S = A × B for some A C X and BCY. 10. (a) Prove that every one-element subset of X x Y is a rectangle. (b) Prove that whenever X is a set with more than one element, the subset {(x, æ) : ¤ € X} S = of X x X is not a rectangle. (Here Y = X.) (c) Is the union of two rectangles in X x Y always a rectangle? Give either a proof that this is true or an example to show that it is false.

Q: Use the element method and contradiction to prove that for all sets S and T S Ռ ( T - S) = {  }

A:

Q: Suppose X and Y are subsets of some universal set. De Morgan's laws provide expressions for the…

A: X∪Yc=Xc∩Yc

Q: Prove or disprove the following. Let A, B, C be sets.   ( A ⊆ B U C ) ⟶  ( A ⊆ B or A ⊆ C )…

A: Consider A,B and C be sets  we have to prove or disprove that ( A ⊆ B U C ) ⟶  ( A ⊆ B or A ⊆ C )…

Q: Let R be the set of real numbers equipped with the standard topology. Let A and B be two subsets of…

A:

Q: Does there exist a non-empty open set A and two elements in A that are farthest apart (among all…

A: There are no two elements that are farthest apart in a non-empty open set. Justification : Let A…

Q: How do I write a set T such that S is an element of T and a proper subset of T?

A:

Q: Let A and B be subsets of a space X such that A is an open set and B is a closed set. Then a. A-B is…

A:

Q: 1. (a) For a set A, P (A) is the set of subsets of A. Write down all the different elements of the…

A: (a)  The empty set, ϕ  is the subset of any set and a set is always a subset of itself. P(ϕ)={ϕ},…

Q: Prove that any closed interval [s, t] can be written as the intersection of a collectable collection…

A:

Q: Show that if S x T is a closed subset of R2, then S and T are closed subsets of R.

A:

Q: Suppose that X is a set and {A. : a E I} is an indexed family of sets (which contains at least one…

A:

Q: Prove that if S is a bounded set then S is bounded.

A:

Q: B. Let B and C be the subsets of some universal set X, then prove that (C – B)' = BU C'

A: let B and C be the subsets of some universet x1 then prove that (c-B)' = BUC'

Q: Problem 21 For each of the following pairs of sets X and Y, compute the symmetric difference of X…

A:

Q: let a be a nonempty compact subset of R and let B be a nonempty closed sub set of R such that A…

A: Let A be a non-empty compact subset of ℝ Let B be a non-empty closed subset of ℝ such that A∩B=∅ To…

Q: If B1,..., Bn is a finite open cover of a compact set A, can the union B1 U... U Bn equal A exactly?…

A:

Q: Determine if the following statements are true or false. Here A represents any set. (a) oCA (b) A'CA…

A: Here let us consider A to be any set which consists some elements U is a universal set  A' is a…

Q: Given a finite set S, find all possible values of |S] so that the set {(a, b) | (a E S) A (b e S) ^…

A:

Q: Prove T={ …,-56,-42,-28,-14,0,14,28,42,…} is a countable set

A: We have to prove the following -   T={ …,-56,-42,-28,-14,0,14,28,42,…} is a countable set.

Q: Prove the following A. Suppose A, B, & C are sets and U the universal set. Show that A - (B . C) =…

A:

Q: Suppose F and G are families of sets. Prove that (UF)n (UG) ≤ U(F N G) iff VA EFVBE G(An BCU (FNG)).

A:

Q: If G(sub alpha) is a collection of closed sets, then the intersections of the G(sub alphas) is a…

A:

Q: 1. Let A, B, and C be sets such that An Bn C + Ø. Draw a Venn diagram for each of the following set…

A:

Q: (c) Prove that only Ø and R are open subsets of R

A: A subset Uof a metric space (M, d) is called open if, for any point x in U, there exists a real…

Q: Cardinality: Let A and B be sets. We write A B (and say A has the same cardinality as B) if there is…

A:

Q: show that empty set, X is an element of S intersection T in topology

A: Here We have to show that empty set must be in S∩T where S and T are topology on set X.

Q: Prove that a set A is countable if and only if there exist an injection of A into N.

A: