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The Venn diagram shows three sets P, Q and R. Use set notation to describe the shaded part.

Mathematics

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Answer

(P∩Q∩R′)∪(P∩R∩Q′)(P \cap Q \cap R') \cup (P \cap R \cap Q')

P∩(QΔR)P \cap (Q \Delta R)

Step-by-step solution

4 steps
  1. Identify the individual shaded regions.

    The shaded region consists of two distinct parts:

    • The part where sets P and Q overlap, but set R does not.
    • The part where sets P and R overlap, but set Q does not.
  2. Express each region using set notation.

    • The region where P and Q overlap is P∩QP \cap Q. If R does not overlap, we exclude R, which is R′R'. So this part is P∩Q∩R′P \cap Q \cap R'.
    • The region where P and R overlap is P∩RP \cap R. If Q does not overlap, we exclude Q, which is Q′Q'. So this part is P∩R∩Q′P \cap R \cap Q'.
  3. Combine the regions using the union operator.

    Since the shaded area includes both of these parts, we use the union (∪\cup) operator:

    (P∩Q∩R′)∪(P∩R∩Q′)(P \cap Q \cap R') \cup (P \cap R \cap Q')
  4. Simplify the expression (optional, but often preferred for conciseness).

    We can factor out P from the expression:

    P∩((Q∩R′)∪(R∩Q′))P \cap ((Q \cap R') \cup (R \cap Q'))

    The expression (Q∩R′)∪(R∩Q′)(Q \cap R') \cup (R \cap Q') represents the symmetric difference between sets Q and R, often denoted as QΔRQ \Delta R.
    Therefore, the shaded region can also be described as:

    P∩(QΔR)P \cap (Q \Delta R)

    The final answer is (P∩Q∩R′)∪(P∩R∩Q′)\boxed{(P \cap Q \cap R') \cup (P \cap R \cap Q')} or P∩(QΔR)\boxed{P \cap (Q \Delta R)}.

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