Combined operations on sets involve performing both intersection and union of sets at the same time. These operations are evaluated from left to right, but any operation enclosed in brackets must always be executed first. This topic also covers finding the complement of combined operations and using Venn diagrams to solve real-world problems.
See how bracketed operations are shaded first, then combined to obtain the final region.
Shade all regions covered by \(P\) or \(Q\).
Shade all regions covered by \(R\).
Shade only the common region between \((P \cup Q)\) and \(R\).
Combined operations on sets involve both intersection \((\cap)\) and union \((\cup)\) simultaneously. Solve from left to right, but always carry out operations inside brackets first.
The complement of a combined set contains all elements in the universal set \((\xi)\) that do not belong to the specified combined set.
Read the question, answer it in your mind, then press “Check Answer”.
If \(\xi = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\), \(A = \{2, 3, 5, 7\}\), and \(B = \{1, 3, 5, 7, 9\}\), what is \((A \cap B)'\)?
Given \(X = \{a, b, c\}\), \(Y = \{b, c, d\}\), and \(Z = \{c, d, e\}\), find the elements of \((X \cup Y) \cap Z\).
In a Venn diagram with sets \(A\), \(B\), and \(C\), which region represents \(A \cap (B \cup C)\)?
The complement of combined operations on sets, and a real-world problem solved step by step.
Carry out the operation inside the brackets first, then take its complement, and finally combine it with the other set.
Given \(\xi = \{x : x \text{ is an integer}, 30 \leqslant x \leqslant 40\}\), set \(A = \{x : x \text{ is a multiple of } 3\}\), set \(B = \{x : x \text{ is a number such that the sum of its two digits is odd}\}\) and set \(C = \{30, 32, 35, 39, 40\}\). List all the elements of \((A \cup B)' \cap C\).
A residents’ association organises sports competitions. 35 participants join the football competition \((A)\), 24 join table tennis \((B)\) and 13 join badminton \((C)\). 4 participants join both football and table tennis, 8 participants join both table tennis and badminton, and 2 participants join all three competitions. No participant joins badminton and football only. Calculate the total number of participants who join one competition only.
Note: the figures 4 and 8 are the totals of those two-set intersections, so they include the 2 participants who join all three. The “two competitions only” regions must therefore have the 2 subtracted first.
The operation \((A \cup B)' \cap C\) means find the complement of \((B' \cap C)\) combined with \(A\).
The prime symbol applies strictly to the expression inside the bracket: first find the union \(A \cup B\), take all elements outside this union in \(\xi\), and finally find the common elements with \(C\).
Complete all parts first, then press the check answer button.
1 Which operation must be carried out first in the expression \(P \cup (Q \cap R)\)?
2 Given universal set \(\xi = \{1, 2, 3, 4, 5\}\), \(A = \{1, 2, 3\}\), and \(B = \{3, 4\}\). What are the elements of \((A \cup B)'\)?
3 Combined operations on sets are solved from left to right, but operations in must be carried out first.
4 For a set \(A\) within the universal set \(\xi\), the complement of set \(A\) is written with the symbol .
Drag each notation to the correct description. On a phone, tap the answer choice first, then tap the description box.
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