Question 11

If $$P, Q, R$$ are subsets of some universal set, then the conditions $$P^c \cap Q \subseteq R^c \cap Q$$ and $$P^c \cap Q^c \subseteq R^c \cap Q^c$$ imply

Solution

$$P^c \cap Q \subseteq R^c \cap Q$$

$$P^c \cap Q^c \subseteq R^c \cap Q^c$$

=> $$\left(P^c\cap Q\right)U\ \left(P^c\cap Q^c\right)\subseteq\left(R^c\cap Q\right)U\ \left(R^c\cap Q^c\right)$$

=> $$P^c\cap\left(Q\ U\ Q^c\right)\subseteq R^c\cap\left(Q\ U\ Q^c\right)$$

=> $$P^c\subseteq R^c$$

=>$$R\subseteq P$$

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