a. Given a set A = (a, b,c) and R defined on A as R = {(a, b), (b, a), (b, b), (c, a)). Explain why R is I. Not Reflexive . Not Symmetric i. Not Anti-symmetric iv. Not Transitive b. What is I R2, . R i. RUR UR c. Hence, show that R UR uR' is transitive.

Q: Prove that in a topological space X, if U is open and C is closed, then U –- C is open and C – U is…

A:

Q: Prove that for any sets A, B, C: (AUB)-B A - B

A:

Q: LAUBUC = ANĒNT. %3D

A:

Q: 4. Prove that, for any sets A and B, it is true that A C B iff AnB = A. Hint. There are two…

A: The intersection is a set operation. The intersection of finite number sets consists of elements…

Q: Given the relation R = {(u,t),(v,t),(w,t),(w,u),(w,v)} on a set A = {t,u,v,w}, what is/are the…

A: topic - relation

Q: If S is compact and D is in S is a closed set, then D is compact. Use the definition of…

A: Let S is compact and D be a closed subset of S. We are to prove that the set D is compact in S. Now…

Q: Let A, B and C be overlapping sets, verify following by using Venn diagram and membership table ((An…

A: Consider the provided question,

Q: 3. Let {B;}je1 and {C;};e1 each be a family of mutually disjoint sets. If B; = C; for each i e I,…

A: Answer :

Q: Let C, D be subsets of a set U. By proving two inclusions, show that (CC n D) U (C n D) = D where Ce…

A:

Q: Let A, B, and C be sets (a) Prove that ACB iff A – B = Ø. (b) Prove that if A C BUCand ANB = Ø, then…

A:

Q: Prove that for sets  A and B,  we have  (A∪B)−(A∩B)=A−B if and only if B⊂A.

A: By theorem we know that,            A⊂B if and only if A∩B =A and  A∪B =B

Q: AAB = (A\ B) U (B\ A). Prove or disprove: (1) (2) (AAB) (AAB)° (AUB) \ (ANB) A¢AB° %3|

A:

Q: Let X = {a, b, c, d}, S = {{a, b, c}, {a}, {b, c, d}}, and T be the topology on X generated by S.…

A:

Q: Let A, B and C be the subsets of U. Then prove that (A UB) – C = (A – C) U(B – C) by using laws of…

A: I have just used the law of sets

Q: Prove that for any sets A, B, C, ANB = AN C and AUB= AUC if and only if B = C. %3D Prove that for…

A:

Q: If A C BỊ U B2 where B1 and B2 are disjoint open sets and A is compact, show that An B1 is compact.…

A: To Show- If A ⊆ B1 ∪ B2 where B1 and B2 are disjoint open sets and A is compact, show that A ∩ B1 is…

Q: Prove formally that A\ B= A\(An B), for any sets A and B.

A:

Q: Let A, B, and C are three sets, prove that C × (AN B)' = (C X A') U (C × B') CS

A:

Q: For any sets A, B,C prove that if BCC then A – (BUC) = A – C. -

A:

Q: - If C is Convex set ond a GR, then XC is a Convex set

A: Let C be a convex set and x∈R Then we have to show that xC is a convex set. The set xC is defined as…

Q: Let A, B, and C be sets. Does A- B= A –C imply B = C? If so, prove it. If not, give a…

A: Let A, B and C are sets. We have to find out that A-B = A- C implies B = C or not If not then we…

Q: 4. By the use of Venn diagrams, in which the space C is the set of points enclosed by a rectangle…

A:

Q: prove that i } X is a Coun table set and P: X>Y is asurjecbive then Y is Countable

A: Note: X denotes cardinality of X, that is number of elements in the set X. If X is countable means…

Q: Let C, D be subsets of a set U. By proving two inclusions, show that (Cºn D) U (C N D) = D where CC…

A: Identities of the sets

Q: Exercise 1. Prove that for sets A, B,C we have C \(AUB) = (C\ A) N (C \ B).

A: We have to show that for sets A, B, C   C ∖(A∪B)=(C∖A)∩(C∖B).

Q: Prove the following theorem directly. 'Let B, E,W be sets. If B Ɔ W, then BU E WUE"

A: Given B, E and W are three sets .

Q: Let (X, T) be a topological space, AcX, then A'next (A) = 10 AO

A:

Q: If X = {a,b, c}, let T1 = {0, X, {a}, {a,b}} and T2 = {0, X, {a}, {b, c}} . Find the smallest…

A:

Q: Let C, D be subsets of a set U. By proving two inclusions, show that (C° n D) U (Cn D)= D where Ce…

A:

Q: Prove that, for any sets A and B, it is true that A ⊆ B iff A ∩ B = A.

A: Use the following definitions, to obtain the solution. Let A and B be any two sets. The…

Q: .et A, B be any two sets. Prove that if A C B, then A n B = p.

A:

Q: Prove the following statements about sets A and B. (AUB)° = A° Bº

A:

Q: Let X = (6,7), and d be the ususal mortric, d(r, y) = |r - yl Gire ezample of a set ACX, to be not…

A:

Q: Suppose A, B, and C are sets, A. Prove that: Ax (Bn C) = (A × B) n (A × C) %3D Using Membership…

A:

Q: Prove that the following statements are EQUIVALENT: For any sets A and B, a. ACB b. AnB = A C. AUB=B

A:

Q: Set C and the universal set U are defined as follows. U={p,q,r,x,y,z} C={p,q,x}

A: Given,        Set C and the iniversal set U are defined as follows :         U=p,q,r,x,y,z…

Q: Let X = {a, b, c, d, e} and let T ={X, Ø, {a}, {c,d}, {a,c,d}, {b,c,d,e} }, then O XT) is neither…

A: Option 1 is correct.

Q: Let U be the universal set, where: U = {a, b, c, d, e, f, g, h, i, j} Let sets A, B, and C be…

A: SOLUTION-

Q: Let A, B and C be the subsets of U. Then prove that (A UB) -C = (A – C)U(B – C) by using laws of…

A: 1) It is the question of computer science.

Q: Given the following sets X= {a,b,c} Y= {a} Z= {a,d} In the universe U= {a,b,c,d,e}, what is (X’…

A: Sample Space (SS): The total number of possible outcomes of an event is called Sample Space and it…

Q: (iii) If (X,r) is topological space, A,BCX. Prove that (a) b(A)=AA (b) If ACB, then d(A) ≤d(B)

A: First part Is obvious from definition of boundary points of a set.

Q: Suppose Ak,., A2 , A1 and B are points on R" Prove that if {A1 , A2, .. Ak} is an independent linear…

A: proof :- A1,A2,...Ak are linear independent set ⇒c1A1+c2A2,...ckAk=0  where c1=c2=...=ck=0…

Q: 1. Prove the following statements about sets A and B. A B= AN Bº %3D

A: Given: Let A and B be two sets. To prove,A-B=A∩Bc

Q: Let U be subset of R". Fill in the blank to define when U is an open set. U is called open if z € U,…

A:

Q: Show that the subset (a, b) of R is homeomorphic with R.

A: Recall that every nonempty open interval (a,b) is homeomorphic to the nonempty interval (c,d).

Q: d) Let X, Y, Z be sets. Calculate |X UYU Z| if |X| = 50, |Y| = 90, |Z| = 71, |X n Y| = 8, |Xn Z| =…

A: Given that X, Y and Z be the three sets.

Q: What is the size of the symmetric difference of two finite sets A and B, that is AA BI ? o IIAI -…

A:

Q: Prove that for all sets A & B, (B-A) = A intersects B^c

A:

Q: If (X,, T,), (X, T2) are two topological spaces, and A,, A, are open sets in X,and X, respectively,…

A:

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.