Question

prove that A union (B intersection c)=(A unionB) intersection (A union c )​

Answer 1

Answer:

It is called "Distributive Property" for sets.Here is the proof for that,

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Let x ∈ A ∪ (B ∩ C). If x ∈ A ∪ (B ∩ C) then x is either in A or in (B and C).

x ∈ A or x ∈ (B and C)

x ∈ A or {x ∈ B and x ∈ C}

{x ∈ A or x ∈ B} and {x ∈ A or x ∈ C}

x ∈ (A or B) and x ∈ (A or C)

x ∈ (A ∪ B) ∩ x ∈ (A ∩ C)

x ∈ (A ∪ B) ∩ (A ∪ C)

x∈ A ∪ (B ∩ C) => x ∈ (A ∪ B) ∩ (A ∪ C)

Therefore,

A ∪ (B ∩ C) ⊂ (A ∪ B) ∩ (A ∪ C).........(1)

Let x ∈ (A ∪ B) ∩ (A ∪ C). If x ∈ (A ∪ B) ∩ (A ∪ C) then x is in (A or B) and x is in (A or C).

x ∈ (A or B) and x ∈ (A or C)

{x ∈ A or x ∈ B} and {x ∈ A or x ∈ C}

x ∈ A or {x ∈ B and x ∈ C}

x ∈ A or {x ∈ (B and C)}

x ∈ A ∪ {x ∈ (B ∩ C)}

x ∈ A ∪ (B ∩ C)

x ∈ (A ∪ B) ∩ (A ∪ C) => x ∈ A ∪ (B ∩ C)

Therefore,

(A ∪ B) ∩ (A ∪ C) ⊂ A ∪ (B ∩ C)..........(2)

So ,

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Hope this will helpfull to u...

Mark as BRILLIANT

Answer 2

Given:

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)​

To prove:

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)​

Solution:

Let x ∈ A ∪ (B ∩ C)

x ∈ A or x ∈ (B and C)

x ∈ A or {x ∈ B and x ∈ C}

{x ∈ A or x ∈ B} and {x ∈ A or x ∈ C}

x ∈ (A or B) and x ∈ (A or C)

x ∈ (A ∪ B) ∩ x ∈ (A ∩ C)

x ∈ (A ∪ B) ∩ (A ∪ C)

x ∈ A ∪ (B ∩ C) -> x ∈ (A ∪ B) ∩ (A ∪ C)

 Hence,

A ∪ (B ∩ C) ⊂ (A ∪ B) ∩ (A ∪ C) - (Equation 1)

Let x ∈ (A ∪ B) ∩ (A ∪ C)

x ∈ (A or B) and x ∈ (A or C)

{x ∈ A or x ∈ B} and {x ∈ A or x ∈ C}

x ∈ A or {x ∈ B and x ∈ C}

x ∈ A or {x ∈ (B and C)}

x ∈ A ∪ {x ∈ (B ∩ C)}

x ∈ A ∪ (B ∩ C)

x ∈ (A ∪ B) ∩ (A ∪ C) -> x ∈ A ∪ (B ∩ C)

Hence,

(A ∪ B) ∩ (A ∪ C) ⊂ A ∪ (B ∩ C) - (Equation 2)

On solving equations 1 and 2,

Since they are subsets of each other they are equal

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)​

Hence proved.