The intersection of sets is an operation that identifies the common elements shared between two or more sets. It is denoted by the mathematical symbol \(\cap\). When examining groups of items, numbers, or survey data, determining the intersection allows us to see exactly where categories overlap, whether represented in set notation, Venn diagrams, or numerical calculations.
See which regions to shade for standard intersections versus their complements.
The overlapping middle region contains elements common to both \(P\) and \(Q\).
Shade everything inside \(\xi\) except the overlapping region.
The central region is common to all three circles.
Only the region where all three circles overlap remains unshaded.
An intersection of sets exists when there is more than one set. The intersection of sets \(P\) and \(Q\) is written using the symbol \(\cap\), denoted as \(P \cap Q\). Set \(P \cap Q\) contains the common elements of both sets \(P\) and \(Q\).
An eclipse of the sun occurs when the moon is between the sun and the earth in a straight line. The intersection can be seen when parts of the moon and the sun occupy the same relative position.
Two complete examples: the number of elements of the complement of an intersection, and a three-set survey problem.
Given the universal set \(\xi = \{x : x \text{ is an integer}, 1 \leqslant x \leqslant 8\}\), set \(A = \{1, 2, 3, 4, 5, 6\}\), set \(B = \{2, 4, 6\}\) and set \(C = \{1, 2, 3, 4\}\). List all the elements and state the number of elements of each set below.
The method: find the intersection first, then list every element of \(\xi\) that is not in that intersection, and finally count them.
A total of 200 university students take part in a survey on the use of technological devices. 155 students have mobile phones \((M)\), 90 students have laptops \((L)\) and 37 students have tablets \((T)\). Of these, 83 students have mobile phones only, 50 students have both mobile phones and laptops only, 5 students have both mobile phones and tablets only, and 4 students have both laptops and tablets only. Calculate the number of students who (a) have all three technological devices, and (b) do not have any of the devices.
Read the question, answer it in your mind, then press “Check Answer”.
Given the universal set \(\xi = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\), set \(M = \{x : x \text{ is an odd number}\}\) and set \(N = \{x : x \text{ is a multiple of } 3\}\). List all the elements of \(M \cap N\).
Given \(\xi = \{x : x \text{ is an integer}, 1 \leqslant x \leqslant 8\}\), set \(A = \{1, 2, 3, 4, 5, 6\}\) and set \(B = \{2, 4, 6\}\). Find the complement of the intersection, \((A \cap B)'\).
In a class of 35 pupils, 15 like to play piano (set \(K\)) and 9 like to play violin (set \(L\)). If 5 pupils like to play both piano and violin, how many pupils do not like to play either musical instrument?
Students often believe that \(P \cap Q\) includes all elements from both sets combined.
That describes the union (\(\cup\)). The intersection (\(\cap\)) includes only elements that are present in both set \(P\) AND set \(Q\) simultaneously.
Believing that if \(B \subset A\), then their intersection cannot be found or is empty.
When \(B\) is a subset of \(A\) (\(B \subset A\)), every element of \(B\) is also inside \(A\), meaning the intersection \(A \cap B\) equals set \(B\).
Complete all parts first, then press the check answer button.
1 Given set \(J = \{1, 3, 5, 7, 9\}\) and set \(K = \{2, 3, 5, 7\}\), what is \(J \cap K\)?
2 Which set notation correctly represents an empty set?
3 If all elements of set \(B\) are found in set \(A\), then set \(B\) is a subset of set \(A\), and \(A \cap B\) is equal to set .
4 The complement of the intersection of sets \(A\) and \(B\) is written using the symbol .
Match each set concept or notation with its correct description. On a phone, tap the answer choice first, then tap the description box.
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