Let E, F and G be finite sets. Let X = (E ∩ F) - (F ∩ G) and Y = (E - (E ∩ G))…

2006

Let E, F and G be finite sets. Let X = (E ∩ F) - (F ∩ G) and Y = (E - (E ∩ G)) - (E - F). Which one of the following is true?

Answer: C. X = Y — Simplify X: X = (E ∩ F) - (F ∩ G) = E ∩ F ∩ G^c. Simplify Y: First, E - (E ∩ G) = E ∩ G^c. Then subtract (E - F) = E ∩ F^c: (E ∩ G^c) - (E ∩ F^c) = E ∩ G^c ∩…

  1. A.

    X ⊂ Y

  2. B.

    X ⊃ Y

  3. C.

    X = Y

  4. D.

    X - Y ≠ φ and Y - X ≠ φ

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Correct answer: C

Simplify X:

X = (E ∩ F) - (F ∩ G) = E ∩ F ∩ G^c.

Simplify Y:

  • First, E - (E ∩ G) = E ∩ G^c.

  • Then subtract (E - F) = E ∩ F^c: (E ∩ G^c) - (E ∩ F^c) = E ∩ G^c ∩ F = E ∩ F ∩ G^c.

Conclusion: X = Y = E ∩ F ∩ G^c.

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