12) Let S1 be a countable set, S2 a set that is not countable, and S1 ⊂ S2.
a) Show that S2 must then contain an infinite number of elements that are not in S1.
b) Show that in fact S2 − S1cannot be countable.
a) S1 is a countable set implies that there exists a bijection from a subset of the natural numbers to S1 and S2 is not countable implies that no bijection exists from any subset of N to S2. Let S3 be the set of elements in S2 that are not in S1. Let S3 be finite. Then S3 and S1 both are countable which implies their union S2 should also be countable. But that is a contradiction. Hence S3 cannot be finite.
b) Let us assume S3 is countable. Since S1 and S3 are both countable, their union S2 should also be countable, which leads to a contradiction. Thus S3, which is the same as S2-S1 cannot be countable.
12) Let S1 be a countable set, S2 a set that is not countable, and S1...
Identify the correct steps involved in proving that the union of a countable number of countable sets is countable. (Check all that apply.) Check All That Apply Since empty sets do not contribute any elements to unions, we can assume that none of the sets in our given countable collection of countable sets is an empty set. If there are no sets in the collection, then the union is empty and therefore countable, Otherwise let the countable sets be As,...
Answer the following questions for the method intersection () below: public Set intersection (Set s1, Set s2) //Effects: If s1 or s2 is null throw NullPointerException /I else return a (non nul) Set equal to the intersection // of Sets s1 and s2 Characteristic: Validity of s1 -s1 has at least one element Characteristic: Relation between s1 and s2 s1 and s2 represent the same set -$1 is a subset of s2 - s2 is a subset of s1 $1...
Let S1 = { 1, 2, 3 }, S2 = { a, b }, S3 = { 4, 5, 6 }. Show a B-tree of minimum degree t = 3 that contains the 18 tuple keys in S1 × S2 × S3, ordered by the linear order defined in (a). Assume that a <2 b in S2. please show the 18 tuple at first which is a cartesian product of s1,s2 and s3 and insert them into a B tree...
q1 1. Consider the alphabet set Σ = (0,1,2) and the enumeration ordering on Σ*, what are the 20th and 25th elements in this ordering? 2. Let N be the set of all natural numbers. Let S1 = { Ag N is infinite }, S2-( A N I A is finite) and S-S1 x S2 For (A1,B1) E S and (A2,B2) E S, define a relation R such that (A1,B1) R (A2,B2) iff A1CA2 and B2CB1. i) Is R a...
The Let s1(t) and s2(t) be defined below: (a) Find an orthonormal basis for S= span{s1(t) and s2(t)}.(b) If y1(t) = 1, find and sketch ý1(t), the projection of y1(t) onto S.
5. Let ф: S1 S2 be a diffeomorphism. a. Show that S is orientable if and only if S2 is orientable (thus, orientability is preserved by diffeomorphisms). b. Let S, and S2 be orientable and oriented. Prove that the diffeomorphism ф induces an orientation in S. Use the antipodal map of the sphere (Exercise 1, Sec. 2-3) to show that this orientation may be distinct (cf. Exercise 4) from the initial one (thus, orientation itself may not be preserved by...
a set (any set of objects) is said to be countable if it is either finite or there is an enumeration (list) of the set. show that the following properties hold for arbitrary countable sets: a) All subsets of countable sets are countable b) any union of a pair of countable sets is countable c) all finite sets are countable
Let X be a set and let T be the family of subsets U of X such that X\U (the complement of U) is at most countable, together with the empty set. a) Prove that T is a topology for X. b) Describe the convergent sequences in X with respect to this topology. Prove that if X is uncountable, then there is a subset S of X whose closure contains points that are not limits of the sequences in S....
3. (8 marks) Let be the set of integers that are not divisible by 3. Prove that is a countable set by finding a bijection between the set and the set of integers , which we know is countable from class. (You need to prove that your function is a bijection.) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Problem 1. Let A C R be a countable set. Prove that R\ A is uncountable.