Answer :

Given: A = {a, b, c, d,}, B = {c, d, e} and C = {d, e, f, g}


(i) Need to prove: A × (B C) = (A × B) (A × C)


Left hand side,


(B C) = {d, e}


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


Right hand side,


(A × B) = {(a, c), (a, d), (a, e), (b, c), (b, d), (b, e), (c, c), (c, d), (c, e), (d, c), (d, d), (d, e)}


(A × C) = {(a, d), (a, e), (a, f), (a, g), (b, d), (b, e), (b, f), (b, g), (c, d), (c, e), (c, f), (c, g), (d, d), (d, e), (d, f), (d, g)}


Now,


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


Here, right hand side and left hand side are equal.


That means, A × (B C) = (A × B) (A × C) [Proved]


(ii) Need to prove: A × (B – C) = (A × B) – (A × C)


Left hand side,


(B – C) = {c}


A × (B – C) = {(a, c), (b, c), (c, c), (d, c)}


Right hand side,


(A × B) = {(a, c), (a, d), (a, e), (b, c), (b, d), (b, e), (c, c), (c, d), (c, e), (d, c), (d, d), (d, e)}


(A × C) = {(a, d), (a, e), (a, f), (a, g), (b, d), (b, e), (b, f), (b, g), (c, d), (c, e), (c, f), (c, g), (d, d), (d, e), (d, f), (d, g)}


Therefore, (A × B) – (A × C) = {(a, c), (b, c), (c, c), (d, c)}


Here, right hand side and left hand side are equal.


That means, A × (B – C) = (A × B) – (A × C) [Proved]


(iii) Need to prove: (A × B) (B × A) = (A B) × (A B)


Left hand side,


(A × B) = {(a, c), (a, d), (a, e), (b, c), (b, d), (b, e), (c, c), (c, d), (c, e), (d, c), (d, d), (d, e)}


(B × A) = {(c, a), (c, b), (c, c), (c, d), (d, a), (d, b), (d, c), (d, d), (e, a), (e, b), (e, c), (e, d)}


Now, (A × B) (B × A) = {(c, c), (c, d), (d, c), (d, d)}


Right hand side,


(A B) = {c, d}


So, (A B) × (A B) = {(c, c), (c, d), (d, c), (d, d)}


Here, right hand side and left hand side are equal.


That means, (A × B) (B × A) = (A B) × (A B) [Proved]


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