Question

If A,B,C are the sets for the letters in the words 'college','marriage','luggage' resp. then verify that A-(BUC)=(A-B)∩(A-C)​

Answer 1

QUESTION:

If A,B,C are the sets for the letters in the words 'college','marriage','luggage' resp. then verify that A-(BUC)=(A-B)∩(A-C)

• So let set A contains the letters of 'college', B for 'marriage', and C for 'luggage'.

A = {c, o, l, e, g}

B = {m, a, r, i, g, e}

C = {l, u, g, a, e}

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First Side of the equation:

• Find the Difference of set A and the union of B and C.

• What is the union of B and C?

⟹ B = {m, a, r, i, g, e}

⟹ C = {l, u, g, a, e}

⟹ BUC = {m, a, r, i, g, e, l, u}

• What is the Difference of Set A and the union of B and C?

⟹ A = {c, o, l, e, g}

⟹ BUC = {m, a, r, i, g, e, l, u}

⟹ A-(BUC) = {c, o}

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Right side of the equation:

• Find the intersection of the difference of A and B and the difference of A and C.

• What is the difference of A and B?

⟹ A = {c, o, l, e, g}

⟹ B = {m, a, r, i, g, e}

⟹ A-B = {c, o}

• What is the difference of A and C?

⟹ A = {c, o, l, e, g}

⟹ C = {l, u, g, a, e}

⟹ A-C = {c, o}

• What is the intersection of the difference of A and B and the difference of A and C?

⟹ (A-B)∩(A-C) = {c, o}

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Conclusion:

$\small \boxed{ \sf \orange{ ∴ A-(BUC)=(A-B)∩(A-C)}}$

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