For a given set S, let P(S) be the collection of all subsets of S. Let binary operations + and on P(S) be defined by A+ B = (AU B) – (An B) and A- B = An B for A, BE P(S). (a) Give the tables for + and for P({x,y}). (b) Prove that for any set S, (P(S), +,) is a ring. (c) Prove or disprove that P({z, y}, +,) is a commutative ring with unity.

Q: (1) (bc' + a'd) (ab' + cd') (ii) b'd + a'bc' + acd + a'bc

A:

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

A:

Q: Show that the space (P is complete.

A: The given problem is to show that the space lp is complete. we can use properties of lp spaces to…

Q: 1. Suppose we would like to define a function f : Q→ Z by the formula f(r) = a +b, where a, b € Z…

A:

Q: Let A= {1,2,x,y,z}, and B= {1,2,3,z}. How many distinct elements are in (A*B) intersect (B*A

A: Given that A=1,2,x,y,zB=1,2,3,z To find the number of distinct elements in A×B∩B×A:

Q: Let S be the set S={(x+y) l 2x+3y>=(greater than or equal to) 12} What would be Sc  (S complement)…

A: The given set is: S=x,y∈ℝ2 | 2x+3y ≥12 Cleary S⊂ℝ2.

Q: Given the function f= {(n,n²)such thatn< 3,n€N} as a set of ordered pairs in set-builder notation,…

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: Let ‘E’ & ‘F’ are any two non-empty sets and there order are as follows;…

A: given let ‘E’ &amp; ‘F’ are any two non-empty sets and there order are as follows;…

Q: Please answer letter D and E onlyyy.

A:

Q: Function f: {a,b)³ → (a,b) is defined by the following rule. For strings € (a,b)³, f(s) = f(x1x2x3)…

A:

Q: Exercise 2.16. Suppose that A and B are countably infinite sets. (A) Show that AUB is countably…

A: Suppose A and B are two countably infinite sets.  To prove that A∪B is also countably infinite.…

Q: F (8)

A: We are given setsWe have to answer (d), (e), (f) and (g).

Q: Let S be the subset of the ℝ2 given by all pairs (x, y) so that x2 + y2 = 2. Show that S is a closed…

A:

Q: Let A be a set and R and S be relations on A. Define the relation R+S on A by R+S = (R – S)U(S – R).…

A: Given A be a set and R and S be relations on A. R+S on A is defined by R+S=R-S∪S-R=R∪S-R∩S=R∪S∩R∩Sc…

Q: What is the kernel and image of the map

A:

Q: Let S be a set defined recursively as follows Basis Step: (1,1) e S. Inductive Step: (a, b) ES (a +…

A: Let S be a set defined recursively as follows, Basis step: 1,1∈SInductive…

Q: 1. (a) Let A = {k, k +1,..., k2}, for k = 1,2,3,.... Uº1 Ak - n=1 Ak (b) Find two infinite sets A…

A: Part (a):- Given that  Ak =  k, k+1, ......, k2 ,   for  k=1, 2, 3,....Find    ∪k=150 Ak - ∩k=1∞ Ak…

Q: Count the number X of functions y: (20, 21,..., 23) → (5,6,...,8) so that is not onto. X =

A: Given function φ:(20,21,...,23)→(5,6,...,8)

Q: Calculate ∪∞n=1 An and ∩∞n=1 Anwhere for each n ∈N the set An is defined as:…

A:

Q: Let S be the subset of the ℝ2 given by all pairs (x, y) so that x2 + y2 = 2. Show that S is a closed…

A: Inverse image of closed set is closed. Whenever f is continous from that we solved this question.

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: There are 16 binary relations on the set {0, 1}: (a) { } (d) {(1,0)} (g) {(0, 0), (1, 0)} (1) {(0,…

A:

Q: (c) Using the set identities, prove that (A – B) n (A – C) = A – (B U C).

A: To Prove: A-B∩A-C=A-B∪C

Q: 3. Express each of these Boolean functions using the operators - and -. a) x+y+z b) x + y(x + 2)

A:

Q: Let X be a set (P(X), Δ,∩) where P(X) is the power set of X, and Δ is the symmetric difference. It…

A: Unity: Let R, +, ∘ be a ring. If the semi-group R, ∘ has identity element then it is called as unity…

Q: Find an orthogonal set with the same span (and same number of elements) as [2] [8]} (18118)} "

A:

Q: Show that for every pair of sets X and Y, the sets X × Y and U {(x, y) | y ≤ Y} xEX are equal.

A:

Q: Choose the correct option. The number of onto function from the set {1,2,….,11} to the set…

A: Let the set A be {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} and the set B be {1, 2, 3, 4, 5, 6, 7, 8, 9,…

Q: Let X be a set with an operation *:X² → X such that (i) x * x = for all x E X, %3D (ii) x * (y * z)…

A:

Q: 5) Let R be a binary relation over the positive integers defined as follows: R = {(a, b) | 2b < a <…

A:

Q: Given the sets X = {1,2,3} and Y = {a,b,c, d}. Define relations a, B, y: X Y by a(1) = b,a(2) = d,…

A: Onto function could be explained by considering two sets, Set A and Set B, which consist of…

Q: Prove: bd (s) = 5 TR-S bd (s) is a closed set

A:

Q: prove: for any two sets X and Y, #(X x Y) = #(X) • #(Y)

A: To prove it equals the product of the cardinality of both the sets.

Q: Suppose a die is cast 10 times. Let X1 be the number of terminations in the set {1, 2}, X2 be the…

A: Let us assume X1 denotes {1,2} faces show in 10 throw . The probability of that will be P(X1=1 and…

Q: It was noted that the binary propositional operators, A and V, can be regarded as functions, f: B x…

A: For any binary operator, number of rows in the truth table =4 Each row can have either True or False…