Question

If A = {a, b, c}, B = {1, 2} and c = {1, 3, e, d}, prove that

A × (B ∪ C) = (A × B) ∪ (A × C)

A × (B ∩ C) = (A × B) ∩ (A × C)​

Answer 1

A = { a,b,c}

B={1,2}

C={1,3,e,d}

1. A×(B U C) = (A ×B) U ( A × C)

B U C = {1,2} U {1,3,e,d}

= {1}

now we have the value of B U C

then we have to find A(B U C)

A×(B U C) = {a,b,c}×{1}

= {a,b,c,1}

second part

(A×B) = {a,b,c}×{1,2}

={a,b,c,1,2}

(A×C) = {a,b,c}×{1,3,e,d}

={ a,b,c,1,3,e,d}

now ,

(A×B) U (A×C) ={a,b,c,1,2} U {a,b,c,1,3e,d}

={a,b,c,1,2}

Answer 2

Answer:

B U C = {1,2,3,e,d}

B ∩ C ={1}

A × B ={(a,1),(a,2),(b,1),(b,2),(c,1),(c,2)}

A × C={(a,1),(a,3),(a,e),(a,d),(b,1),(b,3),(b,e),(b,d),(c,1),(c,3),(c,e),(c,d)}

Now,

A × (B ∪ C)

= {a, b, c} × {1,2,3,e,d}

={(a,1),(a,2),(a,3),(a,e),(a,d),(b,1),(b,2),(b,3),(b,e),(b,d),(c,1),(c,2),(c,3),(c,e),(c,d)}

and

(A × B) ∪ (A × C)

= {(a,1),(a,2),(b,1),(b,2),(c,1),(c,2)} U {(a,1),(a,3),(a,e),(a,d),(b,1),(b,3),(b,e),(b,d),(c,1),(c,3),(c,e),(c,d)

= {(a,1),(a,2),(a,3),(a,e),(a,d),(b,1),(b,2),(b,3),(b,e),(b,d),(c,1),(c,2),(c,3),(c,e),(c,d)}

therefore, A × (B ∪ C) = (A × B) ∪ (A × C)

Now,

A × (B ∩ C)

= {a, b, c} × {1}

={(a,1),(b,1),(c,1)}

and

(A × B) ∩ (A × C)

= {(a,1),(a,2),(b,1),(b,2),(c,1),(c,2)}∩ {(a,1),(a,3),(a,e),(a,d),(b,1),(b,3),(b,e),(b,d),(c,1),(c,3),(c,e),(c,d)}

={(a,1),(b,1),(c,1)}

Therefore, A × (B ∩ C) = (A × B) ∩ (A × C)

Step-by-step explanation:

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