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...

Answer Details

We are given the universal set \(\varepsilon = \{1, 2, 3, \dots, 15\}\) and the three subsets

\[A = \{1, 3, 5, 7, 9, 11\}, \quad B = \{2, 3, 5, 7, 11, 15\}, \quad C = \{3, 6, 9, 12, 15\}.\]

(a) Building the Venn diagram. Work out each region before drawing so that every element sits in exactly one place.

• In all three sets: \(A \cap B \cap C = \{3\}\).
• In \(A\) and \(B\) only: \(\{5, 7, 11\}\).
• In \(A\) and \(C\) only: \(\{9\}\).
• In \(B\) and \(C\) only: \(\{15\}\).
• In \(A\) only: \(\{1\}\).
• In \(B\) only: \(\{2\}\).
• In \(C\) only: \(\{6, 12\}\).
• Outside all three sets: \(\varepsilon \setminus (A \cup B \cup C) = \{4, 8, 10, 13, 14\}\).

These give the Venn diagram below.

(b) Reading answers from the diagram.

(i) \(A'\) is everything outside circle \(A\), so \(C \cap A'\) is the part of circle \(C\) that lies outside \(A\). Reading those regions of \(C\) gives the C-only part \(\{6, 12\}\) together with the \(B\cap C\)-only part \(\{15\}\):

\[C \cap A' = \{6, 12, 15\}.\]

(ii) First form \(B \cup C\) (all of circles \(B\) and \(C\)):

\[B \cup C = \{2, 3, 5, 6, 7, 9, 11, 12, 15\}.\]

Now intersect with \(A'\), keeping only the members of \(B \cup C\) that lie outside circle \(A\). Discarding \(3, 5, 7, 9, 11\) (which are inside \(A\)) leaves

\[A' \cap (B \cup C) = \{2, 6, 12, 15\}.\]