By using closure property

Let ∇ be a binary operation on languages and the class of regular languages is closed under ∇. (∇ can be ∩, ∪, or -)

• If L1 and L2 are regular, then L1 ∇ L2 is regular.
• If L1 ∇ L2 is not regular, then L1 or L2 are not regular.
• If L1 ∇ L2 is not regular but L2 is regular, then L1 is not regular.

Let L={w∈{0,1}*| the number of 0's and 1's in w are equal}.

Let R= {0i1i| i ≥ 0}. R = 0*1* ∩ L

We already prove that R is not regular.

But 0*1* is regular. Then, L is not regular.

Let ∇ be a unary operation on a language and the class of regular languages is closed under ∇.

(∇ can be complement or *)

• If L is regular, then  ∇L is regular.
• If  ∇L is not regular, then L is not regular.

Show that {w∈{0,1}*| the number of 0's and 1's in w are not equal} is not regular

Let L = {w∈{0,1}*| number of 0's and 1's in w are not equal}. Let R ='L = {w∈{0,1}*| number of 0's and 1's in w are equal}. We already prove that R is not regular.

Then, L is not regular.

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