Operations on Regular Languages

TOC: Operations on Regular Languages in Theory of Computation.
Topics Discussed:
1. Union operation on regular languages.
2. Concatenation operation on regular languages.
3. Star operation on regular languages.
4. Theorems on Union and Concatenation.
Full Course on TOC: goo.gl/f4CmJw
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[@starzone9843]

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[@yt_souvik]

I finished this lecture just now. Please upload those two theorem proofs in Another video, it will be helpful for us.

[@omkarmarbhal]

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[@ThePhlox99]

Theorem: The class of regular languages is closed under union and intersection. That
is, if L1 and L2 are regular languages, the so are L1 ∪ L2 and L1 ∩ L2.
Proof Lets start with the union. For simplicity, let us assume that L1 and L2
are languages over the same alphabet Σ.
Since L1 is regular, there exists a DFA
M1 = (Q1, Σ, δ1, q1, F1) which recognizes L1.
Similarly, there exists a DFA M2 =
(Q2, Σ, δ2, q2, F2) which recognizes L2.
To prove that L1∪L2 is regular, we will construct a DFA M∪ which recognizes L1∪L2 =
{w|w ∈ L1 or w ∈ L2}.
The idea: M∪ = (Q, Σ, δ, q0, F) simulates a parallel execution of M1 and M2. M∪ is
defined as follows:
- Q = Q1 × Q2;
- Σ is the same;
- δ((q1, q2), a) = (δ(q1, a), δ(q2, a));
- q0 = (q1, q2);
- F = {(r1, r2) | r1 ∈ F1 or r2 ∈ F2}
To prove correctness we need to show that w ∈ L1 ∪ L2 if and only if M∪ accepts w.
This, in turn, follows from the fact that (r0, r1, . . . , rn) is a computation of M1 on w, and
(t0, t1, . . . , tn) is a computation of M2 on w, if and only if ((r0, t0),(r1, t1), . . . ,(rn, tn))
is a computation of M∪ on w.

[@deepakchand6360]

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[@nitinkymar9468]

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[@sathyasubrahmanyam4166]

Thank u brother

[@MSneberger]

For proof that Regular Languages are closed under Union Operation see Theorem 1.25 on page 45 of Sipser's textbook. For proof that Regular Languages are closed under Concatenation Operation see Theorem 1.26 of Sipser's textbook on page 47.

[@alan2211]

Sipser no make sense.

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[@gedelasivakrishna]

Thankyou !!

[@kasozivincent107]

Hello. I think u messed up the Kleene star operation. It is not just a concatenation of all elements of A. It contains strings of 0 lengths, 1, 2, and so on. So I guess they should even be separated by commas. I suggest it would be better to use union notation. A mathematical definition would suffix for this, if L = {a,b}, L* = {@, a, b, aa, ab, ba, bb, aaa, ...} not just a concatenation of symbols from the alphabet that L is defined. {bbabababbabaabababa} for example is just an element of L*

[@ChoralMasterPiece15]

please upload a video on proofs of the theorms at last

[@mercyoroc8721]

Thank you so much to explaining us 😍😍

[@alitcher4425]

Id like proofs of those 2 theorems please. Thank you for this video as well sir!!!

[@AnonymousDeveloper1]

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Will you record data structures and algorithms tutorial series in the future?

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[@jeetdas4799]

Helps me alot....tnk u sir ji

[@adiyc2909]

Thank you ,sir . Very easily understandable . But can you please provide the proofs of the theorems also ?

[@fnaticbwipo1222]

for proof of the following theorems I will use the fact that regular languages are those that can be described by regular expression, let's say we have 2 regular languages A and B, and their regex be p and q, then concatenation of those languages can be described by pq (first part will describe string from A and second part string from B), as for union we can use | operator for regular expressions and describe language by (p | q) since union contains strings that are present in both languages.

[@shivamrana2130]

thanks bro

[@jaiashutosh2181]

for union purpose dont you think we should mention and like x belongs to a and x belongs to b. please correct me if i am wrong.

[@Ana-el3gk]

Amazing! thank you

[@sasswithdivyanshi]

This is so well explained ❤️❤️❤️❤️❤️❤️❤️

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[@divyaagarwal3091]

Thank you so much sir

[@fcreany9147]

Should the AUB and A•B included the Epsilon ?
Such that
AUB ={ε,pq ,r,t,uv}
A•B={pqt,pquv,rt,ruv, ε,pq(=pqε) ,r,u,t}

[@NSHAIKAMEENA]

No

[@hygieia5672]

For anyone wondering about this in the future, and to be more helpful than just "No", only A* can contain epsilon because in the definition it says there are k elements of A where k can be any positive number including zero. k being zero is how you get epsilon aka no elements at all.

[@Kshirabdi.Tanaya]

Sir please make a video on theorems.🙏🏻

[@dillipkumarsahoo5017]

Sir please make a video of proving the
all theorems of regular languages

[@rumiNITPatna]

thank u sir

[@dulldelusion]

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[@trywellwashington7506]

concatenation: cartesian prod, order of elements in pair immaterial

[@saidurgaprasad5172]

Thanks you sir

[@devmahad]

Union, Concatenation & star.

[@technicalinfochannel66]

dear sir kindly explain the theorem of class of regular language is closed under the "kalena" star operation

[@aydict]

kleene*

[@roshannalawade7389]

sir give me the proof of this

[@muhammadashfaq8032]

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[@jsbbatth8667]

Please make a video on proof of the theorems

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[@dhanushsivajaya1356]

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[@AhamedKabeer-wn1jb]

THANK YOU..

[@pushplataraina51]

hi sir,
i want
proof of theorems

[@adhishagammanpila]

Really appreciate if you could do the proof too

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[@akshunkc6884]

hi sir nice TOC series...plz also upload profs of these theorems...?
thanks in advance...

[@richierabbit2733]

please please make vedios on data structures

[@sourjadhar9332]

The star one is also called kleen closure .

[@uditisinha357]

what is the significance of putting k => 0 for A*?

[@imhsk1037]

pls clarify our doubts

[@UniversalUnicorn]

5:35/5:57 : 🦭🦭

[@nebiyumesfin9620]

Set of regular languages over a given alphabet is not closed under
a) union b) complementation
c) intersection d) none of the above

[@sathirajumadhavarapu3065]

sir, could you please explain proof of two theorems

[@fizzsfizz7617]

Video was awesome but You should provide proof to complete topic

[@Uttamkeshri726]

In A* prq also be there

[@chandanpatra3850]

sir can I use this answer for closure properties of regular sets.

[@knvssandeepbolisetti5008]

Sir pls give the proof for theorems

[@continnum_radhe-radhe]

❤❤❤

[@teenanandwani8823]

Sir please explain Kleen's theorem...

[@kishankanabar6620]

sir i can find one ambiguity over here.When you are declaring definition of star,i.e. A*={x1x2x3...xk:k>=0 and all xi belongs to A}.So when k=0,what would be the concatenating string?and if you are saying it is a null string then again ambiguity arise because as you said all xi,including x0 should belong to A,but that is not in the actual case.Please make it clear.

[@vivekkumar-xc8vq]

Kishan Kanabar Hey... Are you not aware of the fact that all sets have a null element i.e. null element belongs to each and every set....

[@risingsingh3424]

How to visualise practically the concept of regular languages

[@satwikyalamarthi1162]

We want proof...... Can you please explain

[@RNCSSakshamBhatia]

If possible please give the proof of theorems.

[@Ankit-we8ym]

dear sir are you also teaching any other subjects for cse students

[@nitishbharat9942]

Yes 👍

[@moazzamali5995]

Sir plz..show some prove. Of this theorem

[@Lavnder-k7x]

Proof please sir

[@akshaykhachi8565]

If L={ab, c}...what will be L^3....??

[@Montserratx]

Can you explain to me this example, please?
A*={a*,b*,a*,b*}

[@durgarajeswaritalluri4124]

sir will pqpqr is also in A* ??

[@mantashawasi1991]

Yes

[@gajulanagamani8312]

sir explaine the proof about two theorems

[@saptarshiroy3943]

Sir, can you provide the proof of Regular language expression?

[@ishmitabasu9623]

Can i please get a proof of the two theoram?

[@saraswatibepari8233]

Sir , give us the prove of this 2 theorem ... please

[@ronikseth6742]

sir i want regular language's many types of problem with solution for better understand.thank you.

[@thirumenipharma2612]

what about (A+B)*

[@CyrusXIII]

Thank you :)

[@shankarreddy4674]

What is it mean every one commenting like this

[@Amartya1607]

theorem proofs?

[@bhishmapitamah3681]

5:52 your device is fine

[@ashisraha9263]

Sir I need proves of two theorem of regular languages. If you send me then I will be thankful to you.

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Sir please do proo

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Sir please give the proof of the theorem.

[@barathkumarsr9890]

Please prove the theorem 1& 2 sir

[@pedarayududola]

what is regular language

[@sambhavjain9382]

can we have the proofs please?

[@pranaysahariah6193]

Proof of the therom sir pls

[@msm8416]

Proof for Theorem 1,Theorem 2 plz

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sir proof please on union operation

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thermos of regular languages expain sir

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Proof please

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Want the proof