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Question
Solve the following :
If A = `[(1, 5),(7, 8),(9, 5)], "B" = [(2, 4),(1, 5),(-8, 6)] "C" = [(-2, 3),(1, -5),(7, 8)]` then show that (A + B) + C = A + (B + C).
Solution
(A + B) + C = `{[(1, 5),(7, 8),(9, 5)] + [(2, 4),(1, 5),(-8, 6)]} + [(-2, 3),(1, -5),(7, 8)]`
= `[(1 + 2, 5 + 4),(7 + 1, 8 + 5),(9 - 8, 5 + 6)] + [(-2, 3),(1, -5),(7, 8)]`
= `[(3, 9),(8, 13),(1, 11)] + [(-2, 3),(1, -5),(7, 8)]`
= `[(3 - 2, 9 + 3),(8 + 1, 13 - 5),(1 + 7, 11 + 8)]`
∴ (A + B) + C = `[(1, 12),(9, 8),(8, 19)]` ...(i)
A + (B + C) = `[(1, 5),(7, 8),(9, 5)] + {[(2, 4),(1, 5),(-8, 6)] + [(-2, 3),(1, -5),(7, 8)]}`
= `[(1, 5),(7, 8),(9, 5)] + [(2 - 2, 4 + 3),(1 + 1, 5 - 5),(-8 + 7, 6 + 8)]`
= `[(1, 5),(7, 8),(9, 5)] + [(0, 7),(2, 0),(-1, 14)]`
= `[(1 + 0, 5 + 7),(7 + 2, 8 + 0),(9 - 1, 5 + 14)]`
∴ A + (B + C)= `[(1, 12),(9, 8),(8, 19)]` ...(ii)
From (i) and (ii), we get
(A + B) + C = A + (B + C).
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