someone pls help to proof these expressions

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Idempotent means if operands of your operator are equal then you should get operand as o/p

V) a+a =a

you know that a+b means either a or b ===> put a in place of b ===> a+a = either a or a = a ===> Follow idempotent Law

VI) a.a ҂ a

you know that a.b means a concatenate with b ===> put a in place of b ===> a.a = aa ҂ a ===> Doesn't Follow idempotent Law

I) (P+Q)* P* Q* ------------> every string in the language, actually you can get every string in the language by (P+Q)*

therefore (P+Q)* P* Q* = (P+Q)*

II) can be self explanatory after getting (I)

III) ∊ + R = either ∊ or R, if R contains ∊ ===> ∊ + R = R

IV) ∊ + R = R? -----------------> False

R? means R can be one time or zero

ex:- R=ab

∊ + R = { ∊ , ab}

R? = {∮, ab}

V) a+a =a

you know that a+b means either a or b ===> put a in place of b ===> a+a = either a or a = a ===> Follow idempotent Law

VI) a.a ҂ a

you know that a.b means a concatenate with b ===> put a in place of b ===> a.a = aa ҂ a ===> Doesn't Follow idempotent Law

I) (P+Q)* P* Q* ------------> every string in the language, actually you can get every string in the language by (P+Q)*

therefore (P+Q)* P* Q* = (P+Q)*

II) can be self explanatory after getting (I)

III) ∊ + R = either ∊ or R, if R contains ∊ ===> ∊ + R = R

IV) ∊ + R = R? -----------------> False

R? means R can be one time or zero

ex:- R=ab

∊ + R = { ∊ , ab}

R? = {∮, ab}

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