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# Explain the technique for combining two languages in TOC?

There are three common ways of creating a new language from two languages −

Union

Intersection

Product

Languages are sets of strings, so they can be combined by the usual set operations of union and intersection.

**Intersection**

If L1 and L2 are languages over ∑, then L1 ∩ L2 is the language of strings in both L1 and L2 .

For example, If L = {aa, bb, ab} and M = {ab, aabb} then,

The **intersection** is as follows −

L ∩ M = {ab}

And

**Union**

If L1 and L2 are languages over the alphabet ∑, then the language L1 ∪ L2 is the language of all strings in at least one of the two languages.

For example, If L = {aa, bb, ab} and M = {ab, aabb} then,

The **union** is as follows −

L ∪ M = {aa, bb, ab, aabb}.

Where L is M are two languages over an alphabet ∑.

**Concatenation** − The operation of concatenation of strings places two strings in juxtaposition.

The concatenation of two languages L1 and L2 over the alphabet ∑ is the language L_{1}L_{2} = { wx ∈ ∑* | w ∈ L_{1} ∧ x ∈ L_{2} }

The set of strings that can be split into two pieces − a piece from L_{1} and a piece from L_{2}. It is similar to the Cartesian product of two sets, with strings.

**Example**

The concatenation of the two strings aab and ba is the string aabba.

We use the name cat to denote this operation − cat(aab, ba) = aabba.

We can combine two languages L and M by forming the set of all concatenations of strings in L with strings in M, which is called the product of the two languages.

**Definition (Product of two languages)**

If L and M are languages, then the new language called the product of L and M is defined as L · M (or only LM)

L · M = {cat(s,t) : s ∈ L and t ∈ M}

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