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Und LoJicfor which we may not know- els Exenise set?"' tor every object _ 7. (0) Are there well-defined element of thie question AIs< which we do not k...

Question

Und LoJicfor which we may not know- els Exenise set?"' tor every object _ 7. (0) Are there well-defined element of thie question AIs< which we do not know the answcr for - cardinality? Are there finite sets in Exercise h.r; e}. and so 8. Let $; =d o} Ss = {t, e}. $z IO} with ISl =4 Find all k € (1.2. N with $; = St Find distinct indices j. k Find the smallest value of k € N with whether the following statements are tue Let 9 = {Silie,: Determine false. ix. Si $ Sz1 Su = {n.e.i,

und LoJic for which we may not know- els Exenise set?"' tor every object _ 7. (0) Are there well-defined element of thie question AIs< which we do not know the answcr for - cardinality? Are there finite sets in Exercise h.r; e}. and so 8. Let $; =d o} Ss = {t, e}. $z IO} with ISl =4 Find all k € (1.2. N with $; = St Find distinct indices j. k Find the smallest value of k € N with whether the following statements are tue Let 9 = {Silie,: Determine false. ix. Si $ Sz1 Su = {n.e.i,t,h,e,r} S1 C S21 {n, e.t} < Sz0 iii. Si € $ {n,i,e} e $ iv S; 5 $ xii ({f,o,u,r}} E9 0 e y xiii: u € 540 vi: 0 € y xiv 9 (S9) < 9(S19) vii: 0 6 $ {s,i} e %(S6) viii Si € Si, xvi: W € 9(S2) 9. For k € {1,2,... 20}, let Dt = {xIx is & prime number that divides k} 2 = {Delk e (1,2, and Iat 20}}. Find D . Dz, D1o, and Dzo. True or false: Dz € D1o vii: {5} € 9 i Da€ Do viii. {4,5} e 9 Dio € Dzo iv: @e 9 {{3}} < 9 0 € 9 9(Do) @(D6) vi S€ 9 xi 9(3,4}) = 9 (C) Find |Dvol and |Dtgl: xii. {2,3} € 9(D1z) (d) Find |91. Give an example of an indexed = collection 9 = {S;}}_1 with |S| =3 Operations are some standard set operations - used to derive new sets from



Answers

Construct non deterministic finite-state automata that recognize each of the sets in Exercise 8 .

All right, So you want the cardinality of these sets and so we can do this because he's bigger than it. So we can take B minus, which is just equal to lowercase B when it's focusing.

All right. So we want the size of uh may I speak? So we know that A lives inside of me. So this is A. This is me. And there's no way that um we can take all of be away right? Without and keep A. So if I take all the b away, then I have take care way too, because they're inside inside of people. So this is just another way of saying that this is equal to just the cardinality of the empty set, right? If I take everything away, that means I have nothing left. So that's the empty set. And the cardinality of the empty set is just zero.

Once again, welcome to a new problem. When you think about sets, you can think about the cardinality cardinality of sets. And this is pretty much uh a measure measure a measure of the, of the number of elements. Number of elements in a set. Uh a measure of the number of elements in the set. For example, if I have a set A 1, 2, 3, 4 then the card analogy of this set would be the same as for meaning. It's the number of elements in this set. Um is four. And then in this particular problem we want to determine if we want to determine the cardinality, the cardinality of uh these sets. We want to find out the cardinality of these sets. So in part a mm hmm we have that set and then uh and the second part would be. Mm hmm. But b we have another set. So this is the second set for part B. Mm hmm. And then in but see We have this one and then in the body we of neither one of these sets. So we want to figure out what's the cardinality of these sets that you're looking for. So in this sense was saying uh we're counting the number of elements. So I will say the set was one element and cardiology. Mm hmm. One. And then in part B will say uh the search as one element and cardinality. Mm hmm One. And then in part C we're gonna say uh this set mm hmm has to relevance and cardinality two. And then the last one. But we were going to say that, mm hmm. The set. It has three elements and cardinality three weeks. So hope you enjoyed the problem. Feel free to send any questions or comments and have a wonderful day.

Were given sets were asked to construct non deterministic funded state of Ramadan. Recognize each of these sets. Sets come from previous exercise exercise eight. So, in part A, we're given the set, which is the empty set. So this consistent No strings. Now, since this is a finance state, atomic don, you must have a start state. So let s zero be a start state because the empty string should not be recognized. It follows that as zero should be a non final state. And from this we obtain in non deterministic financed. Your Thomason does not recognize any strings how in part B were given the sec consisting of the empty string. Once again, we'll let s zero be a start state now because the machine should recognize the empty string it follows, that s zero should be a final state. However, the machine shouldn't recognize any other strings. So this is all that we should have. So the resulting non deterministic finance data Tom it on will only recognize the empty strength cause there can be noble transitions with inputs. Now, in part C, we're given the set consisting of the letter A where a is a letter in some alphabet I once again, but at zero be a start state because the empty string should not be accepted. It follows That s zero should be a non final state. Now suppose that the input is a Then we're going to move from S zero to final state s one since a should be accepted by the machine.


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