The sets A = {1, 3, 5, 7, 9, 11}, B = {2, 3, 5, 7, 11, 15} and C = {3, 6, 9, 12, 15} are subsets of \(\varepsilon\) = {1, 2, 3, ..., 15}. (a) Draw a Venn di...
Question 1 Report
The sets A = {1, 3, 5, 7, 9, 11}, B = {2, 3, 5, 7, 11, 15} and C = {3, 6, 9, 12, 15} are subsets of \(\varepsilon\) = {1, 2, 3, ..., 15}.
(a) Draw a Venn diagram to illustrate the given information.
(b) Use your diagram to find : (i) \(C \cap A'\) ; (ii) \(A' \cap (B \cup C)\).
Answer Details
(a) Venn diagram
(b)(i) \(C \cap A'\) means the elements that are in \(C\) but not in \(A\).
From the diagram, these are the \(C\)-only region \(\{6,12\}\) and the \(B \cap C\) only region \(\{15\}\):
\[C \cap A'=\{6,12,15\}.\]
The supplied reference answer \(\{15\}\) is not consistent with the stated sets: \(6\) and \(12\) are both in \(C\) and neither is in \(A\), so they must be included.
(b)(ii) First identify all elements in \(B\) or \(C\):
\[B\cup C=\{2,3,5,6,7,9,11,12,15\}.\]
Intersecting with \(A'\) removes every element that is in \(A\), namely \(3,5,7,9,11\). Therefore:
\[A'\cap(B\cup C)=\{2,6,12,15\}.\]
Examination reminder: An intersection with \(A'\) keeps only elements outside \(A\). Check each region of the Venn diagram that lies outside the \(A\) circle.
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