Intersection of two sets

Part ofMathsSets and Venn diagrams

Key points

An image of a Venn diagram. The Venn diagram has been used to classify shapes. One circle represents, squares, the other circle represents, blue shapes.
The intersection contains blue squares, which are elements common to both sets 𝑨 and 𝑩.

  • An of a is a member of that set.

  • An element that is contained in two different sets is a member of both of those sets and is in the

  • The intersection contains any elements that are common to all of the sets being considered.

  • The of two or more numbers can be found using the intersection of sets in a .

An image of a Venn diagram. The Venn diagram has been used to classify shapes. One circle represents, squares, the other circle represents, blue shapes.
The intersection contains blue squares, which are elements common to both sets 𝑨 and 𝑩.
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Understanding and using the intersection of sets

  • The intersection of two or more sets is the overlap between the sets.

  • The symbol ∩ is used to show that sets .

  • \(A\)∩\(B\) is the intersection of set \(A\) and set \(B\) and contains the set of elements which are in both set \(A\) and set \(B\).

  • \(A\)∩\(B\)∩\(C\) is the intersection between set \(A\), set \(B\) and set \(C\) and contains the set of elements which are in set \(A\), set \(B\) and set \(C\).

  • The symbol \('\) is written after the lettered name of the set to give the complement of the set. \(Q'\) is the complement of set \(Q\).

Examples

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  1. A series of two images. Each image shows the same Venn diagram with two intersecting circles. The circle on the left is labelled, A. The circle on the right is labelled, B. A rectangle has been drawn around the outside of the two circles. Outside the rectangle, in the top left: the symbol, ξ. The overlap between the two circles, or intersection, is coloured orange. Written below the first Venn diagram: intersection. Written below the second Venn diagram: A, intersection symbol, B.

    The intersection of a set is the overlap between the sets. 𝑨∩𝑩 is the intersection of set 𝑨 and set 𝑩.

Question

List the elements in 𝑨∩𝑩 and describe the intersection.

An image of a Venn diagram with two intersecting circles. The circle on the left, is labelled, A. The circle on the right, is labelled, B. A rectangle has been drawn around the outside of the two circles. Outside the rectangle, in the top left: the symbol, ξ. Written above: ξ equals, open brace bracket, one, comma, two, comma, three, comma, four, comma, five, comma, dot, dot, dot, twenty, close brace bracket. A equals, open brace bracket, square numbers, close brace bracket. B equals, open brace bracket, odd numbers, close brace bracket.

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Use a Venn diagram to find the HCF of two or more numbers

  1. Write each number as a without using .
  2. Draw a circle for each number. The circles will overlap.
  3. Place the factors that are common to both (or all) numbers in the of the circles and place the remaining prime factors for each number in its own circle.
  4. Multiply the numbers in the intersection. The product of these numbers is the HCF.

Examples

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  1. Example one. The H C F of two numbers. An image of a Venn diagram with two intersecting circles. The circle on the left is labelled, forty. The circle on the right is labelled, forty eight. A rectangle has been drawn around the outside of the two circles. Outside the rectangle, in the top left: the symbol, ξ.

    Find the highest common factor (HCF) of 40 and 48

Question

Given the product of prime factors of 36 and 54, use the Venn diagram to work out the highest common factor of 36 and 54.

An image of a Venn diagram with two intersecting circles. The circle on the left is labelled, thirty six. The circle on the right is labelled, fifty four. A rectangle has been drawn around the outside of the two circles. Outside the rectangle, in the top left: the symbol, ξ. Written above: thirty six equals two multiplied by two multiplied by three multiplied by three. Fifty four equals two multiplied by three multiplied by three multiplied by three.

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Practise understanding and using the intersection of two sets

Quiz

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