Question

$\large{\bf\purple{Hᴇʏ}}!!$

$\huge\purple{\underbrace\pink{ \: \: ǭєรԵเ๏ภ \: \: }}$
Lᴇᴛ A  ᴀɴᴅ B ʙᴇ sᴇᴛs ; if A∩X = B∩X = ∅ ᴀɴᴅ A∪X = B∪X ғᴏʀ sᴏᴍᴇ sᴇᴛ X.Sʜᴏᴡ ᴛʜᴀᴛ A=B..

ƘíղժӀվ ղօ ՏԹɑʍ.
ภєє๔ ฬєll є×קlคเภє๔ ɑภรฬєг.
​

Answer 1

Answer:

A ∩ X = B ∩ X = ∅

A ∪ X = B ∪ X

Let,

A = A ∩ (A ∪ X)

⇒ A = A ∩ (B ∪ X)

    ∵ A ∪ X = B ∪ X

By distributive law,

A ∩ (B ∪ X) = (A ∩ B)∪(A ∩ X)

                   =  (A ∩ B) ∪ ∅

       (A ∩ X) = B ∩ X = ∅

∴ A = A ∩ B  ____ (1)

Let,

 B = B ∩ (B ∪ X)

⇒ B = B ∩ (A ∪ X)

   ∵ A ∪ X = B ∪ X

Again by distributive law,

B ∩ (A ∪ X) = (B ∩ A)∪(B ∩ X)

                   =  (B ∩ A) ∪ ∅

       (A ∩ X) = B ∩ X = ∅

∴ B = (B ∩ A)

⇒ B = (A ∩ B) ____ (2)

From equation (1) and (2), we have

A = (A ∩ B) = B

⇒ A = B

Hence Proved !!

Answer 2

Answer:

A ∩ X = B ∩ X = ∅

A ∪ X = B ∪ X

Let,

A = A ∩ (A ∪ X)

⇒ A = A ∩ (B ∪ X)

    ∵ A ∪ X = B ∪ X

By distributive law,

A ∩ (B ∪ X) = (A ∩ B)∪(A ∩ X)

                   =  (A ∩ B) ∪ ∅

       (A ∩ X) = B ∩ X = ∅

∴ A = A ∩ B  ____ (1)

Let,

 B = B ∩ (B ∪ X)

⇒ B = B ∩ (A ∪ X)

   ∵ A ∪ X = B ∪ X

Again by distributive law,

B ∩ (A ∪ X) = (B ∩ A)∪(B ∩ X)

                   =  (B ∩ A) ∪ ∅

       (A ∩ X) = B ∩ X = ∅

∴ B = (B ∩ A)

⇒ B = (A ∩ B) ____ (2)

From equation (1) and (2), we have

A = (A ∩ B) = B

⇒ A = B

Hence Proved !!

Step-by-step explanation:

tq for thanks.mere friend banoge ap