Below are four statements about three non-empty sets $P$, $Q$, and $R$ in a universal set $U$ - Junior Cycle Mathematics - Question 13 - 2018
Question 13
Below are four statements about three non-empty sets $P$, $Q$, and $R$ in a universal set $U$.
(a)
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Worked Solution & Example Answer:Below are four statements about three non-empty sets $P$, $Q$, and $R$ in a universal set $U$ - Junior Cycle Mathematics - Question 13 - 2018
Step 1
1. $\#(P \cup Q) = \#(P) + \#(Q)$
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Answer
This statement is sometimes true. While it's true for disjoint sets (where P∩Q=∅), it is not always true when sets overlap. For overlapping sets, the correct relation should reflect the intersection: #(P∪Q)=#(P)+#(Q)−#(P∩Q).
Step 2
2. $P \cap Q = P \cap Q$
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Answer
This statement is never true. This is essentially a tautology; it cannot be true as it equates itself without giving any new information or significance. Therefore, it doesn't give a true representation of the relationship between the sets.
Step 3
3. $(P \cap Q)' = P' \cup Q'$
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Answer
This statement is always true. This is a direct application of De Morgan's Laws in set theory, which states that the complement of the intersection of two sets is equal to the union of their complements.
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Answer
This statement is always true. This also follows from the distributive property of set operations, demonstrating how unions distribute over intersections.
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