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- 27. Let , where and are nonempty. Prove that has the property that for every subset of if and only if is one-to-one. (Compare with Exercise 15 b.). 15. b. For the mapping , show that if , then .For the given f:ZZ, decide whether f is onto and whether it is one-to-one. Prove that your decisions are correct. a. f(x)={ x2ifxiseven0ifxisodd b. f(x)={ 0ifxiseven2xifxisodd c. f(x)={ 2x+1ifxisevenx+12ifxisodd d. f(x)={ x2ifxisevenx32ifxisodd e. f(x)={ 3xifxiseven2xifxisodd f. f(x)={ 2x1ifxiseven2xifxisoddDescribe the kernel of epimorphism in Exercise 20. Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].
- Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].Let (X,T) be a topological space, K a compact subset of X and (Fn)n∈ N a family in P(X) with Fn≠ø for all n. If K⊇F̅0⊇F̅1⊇....⊇F̅n⊇........ then prove that (by way of contradiction) n=0∩∞ F̅n ≠øLet (X, B) be a measurable space and {μn} a sequence ofmeasures with the property that for every E ∈ B,μn(E) ≤ μn+1(E), n = 1, 2,... .Let μ(E) = limn→∞ μn(E). Show that (X, B, μ) is a measurespace.
- Give an example of a set X and topologies T1 and T2 on X such that T1 union T2 is not a topology on XLet (X,d) be a metric space , x ϵ X and A ⊑ X be a nonempy set. Prove that d (x ,A) = 0 if and only if every neighborhood of x contains a point of A.d((x1, x2, x3), (y1, y2, y3) = |x1 - y1| + |x2 - y2| + |x3 - y3|. Conclude that, with this metric, a subset of ℝ3 is sequentially compact if and only if it is closed and bounded.
- Let (+) and (Y ,A) be two topological spaces. Let # be a base for A. Prove that a function f :(4’.r)——({Y ,A) is continuous if and only if the inverse image under ¢ of every member of / is a r- open SetLet (S,≼) be a linearly ordered set with the Least-Upper-Bound Property. Let A and B be non-empty and bounded below subsets of S. (i) Prove that A ∪ B is bounded below in S.(ii) Prove that inf(A ∪ B) ≼ inf(A) by using the definition of the infimum of a set in S.Consider the space Z+ with the finite complement topology. Consider the sequence (xn) of points in Z+ given by xn = n+7. To what point or points does the sequence converge?