- Trending Categories
- Data Structure
- Networking
- RDBMS
- Operating System
- Java
- iOS
- HTML
- CSS
- Android
- Python
- C Programming
- C++
- C#
- MongoDB
- MySQL
- Javascript
- PHP

- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who

# What are the basic properties of products in TOC?

It is easy to see that for any language L the following simple properties hold −

L · {∧} = {∧} · L = L

L · ∅ = ∅ · L = ∅

Now let’s see the commutativity and associativity of the operation of concatenation.

**Properties of products – commutativity**

The operation of concatenation is not commutative. In other words, the order matters!

Given two languages L and M, it’s usually true that

L · M ≠ M · L

**Example**

If L = {ab, ac} and M = {a, bc, abc}, then the product

L · M is the language

L · M = {aba, abbc, ababc, aca, acbc, acabc},

But the product M · L is the language

M · L = {aab, aac, bcab, bcac, abcab, abcac}

These have no strings in common!

**Properties of products – associativity**

The operation of concatenation is associative. In other words, if L, M, and N are languages, then

L · (M · N) = (L · M) · N

**Example**

If L = {a, b}, M = {a, aa} and N = {c, cd}, then

L · (M · N) = L · {ac, acd, aac, aacd}

= {aac, aacd, aaac, aaacd, bac, bacd, baac, baacd}.

This is the same as,

(L · M) · N = {aa, aaa, ba, baa} · N

= {aac, aacd, aaac, aaacd, bac, bacd, baac, baacd}.

- Related Questions & Answers
- What are the properties of Regular expressions in TOC?
- Explain the basic properties of the Turing machine?
- what are the latest apple products?
- What are the fundamental concepts of TOC?
- What products of Patanjali are worth buying?
- What are the undecidable problems in TOC?
- What are the Basic Principles of Light Control?
- What are the basic concepts of data mining?
- What are the basic computational models?
- What are the Turing machine variations in TOC?
- What are the properties of MySQL user variables?
- What are the closure properties of Regular languages?
- What are the Properties of Data Encryption Standard?
- What are the properties of array class in C#?
- What are the properties of window.screen object in JavaScript?