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CFL refers to Context Free Language in the theory of computation (TOC). Let us now understand how CFL is closed under Union.

**CFL is closed under UNION**

If L1 and L2 are CFL’s then L1 U L2 is also CFL.

Let L1 and L2 are generated by the Context Free Grammar (CFG).

G1=(V1,T1,P1,S1) and G2=(V2,T2,P2,S2) without loss of generality subscript each non terminal of G1 and a1 and each non terminal of G2 with a2 (so that V1∩V2=φ).

Subsequent steps are used production entirely from G1 or from G2.

Each word generated thus is either a word in L1 or L2.

**Example**

Let L1 be palindrome, defined by:

S->aSa|bSb|a|b|^

Let L2 be {anbn|n>=0} defined by −

S->aSb|^

Then the union language is defined as −

S->S1|S2

S1->aS1a|bS1b|a|b|^

S2->aS2b|^

**CFG are closed under KLEENE star**

Now, let us understand how CFG is closed under a star.

**Proof**

If L1 is a CFL then L1* is a CFL.

Let L1 be generated by CFG G1=(V1,T1,P1,S1) without loss of generality subscript each non-terminal of G1 with a1.

Define the CFG, G generates L1* as −

G=(v1 U {S}, T1, P1 U {S->S1S{^},S}

Each word generated either ^ or the same sequence of words in L1.

Every word in L1* can be generated by G.

**Example**

Let L1 be {anbn|n>=0} defined by S->aSb|^

Then L1* generated as:

S->S1S|^

S1->aS1b|^

**CFGs are not closed under intersection**

Let us now understand how CFG cannot be closed under intersection.

**Proof**

If L1 and L2 are CFL’s then L1∩L2 may not be a CFL.

L1={anbnam|n,m>=0} is generated by the CFG −

S->XA

X->aXb|^

A->Aa|^

L2-{anbmam|n,n>=0} is generated by CFG −

S->AX

X->aXb|^

A->Aa|^

L1∩L2 ={anbnan|n>=0} which is not to be a CFL.

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