Union of two sets

Part ofMathsSets and Venn diagrams

Key points

An image of a Venn diagram being used to classify integers based on their properties. The union of the Venn diagram has been shaded orange.
𝑷ᴜ𝑸 is the union of set 𝑷 and set 𝑸.

  • The combination of two or more is the of the sets.

  • The for union is ᴜ.

  • For example, the union of set \(A\) and set \(B\) is written as \(A\)ᴜ\(B\). The union of set \(X\), set \(Y\) and set \(Z\) is written as \(X\)ᴜ\(Y\)ᴜ\(Z\).

  • The union contains the that are in each set and any elements that are common to the sets being considered.

  • The lowest common multiple (LCM) of two or more numbers can be found using the union of sets. For this it is important to know about common factors.

An image of a Venn diagram being used to classify integers based on their properties. The union of the Venn diagram has been shaded orange.
𝑷ᴜ𝑸 is the union of set 𝑷 and set 𝑸.
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Using the union of two sets

  • To list the elements in \(A\)ᴜ\(B\), list those that are:

    • In both set \(A\) and set \(B\)
    • In only set \(A\)
    • In only set \(B\)

  • The elements can be listed in any order, and starting with the elements can help in avoiding duplication of elements.

  • To identify \(A\)ᴜ\(B\) on a Venn diagram:

    • Use the that is made up of the combined set circles for set \(A\) and set \(B\).

  • Remember that all the data being considered is contained inside the rectangle. This is called the and is represented by the Greek letter ξ (Xi). Each circle represents a .

Examples

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  1. Example one. Listing the elements in the union of two sets. The sets: ξ equals, open brace bracket, a series of nine images, close brace bracket. Each image is a two dimensional shape. The first shape is a green equilateral triangle. The second shape is a purple pentagon. The third shape is a blue equilateral triangle. The forth shape is a green square. The fifth shape is a purple equilateral triangle. The sixth shape is a green pentagon. The seventh shape is a blue square. The eighth shape is a green right angled triangle. The ninth shape is a green circle. Written below: A equals, open brace bracket, triangles, close brace bracket. Written beneath: B equals, open brace bracket, green shapes, close brace bracket.

    List the elements of 𝑨ᴜ𝑩, the union of set 𝑨 and set 𝑩.

Question

The Venn diagram shows the number of people in a large office who bring home-baked treats into the office to share.

How many people do this?

An image of a Venn diagram with three intersecting circles. The circle on the left, is labelled, Cakes. The circle on the right, is labelled, Biscuits. The circle below, is labelled, Pastries. A rectangle has been drawn around the outside of the three circles. Outside the rectangle, in the top left: the symbol, ξ. The Venn diagram has been populated with numbers. The number, eight, is in the intersection of all three circles. The number, six, is in the other part of the intersection for the circles labelled Cakes and Biscuits. The number, seven, is in the other part of the intersection for the circles labelled Biscuits and Pastries. The number, nine, is in the other part of the intersection for the circles labelled Cakes and Pastries. The number, fifteen, is in the region of the circle labelled, Cakes, that does not overlap with any other circle. The number, four, is in the region of the circle labelled, Biscuits, that does not overlap with any other circle. The number, three, is in the region of the circle labelled, Pastries, that does not overlap with any other circle. The number, eighteen, is outside the circles, in the rectangular box.

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Finding the lowest common multiple using the union of sets

  • To use a Venn diagram to find the lowest common multiple (LCM) of two numbers:
  1. Write each number as a without using .
  2. Draw a circle for each number. The circles will overlap.
  3. Place the in the of the circles and place the remaining prime factors for each number in its own circle.
  4. Multiply the numbers in the of the sets. The product of all these numbers is the LCM.

Example

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  1. Example. Finding the L C M of two integers, whole numbers. Seventy equals two multiplied by five multiplied by seven. Written below: eight equals two multiplied by two multiplied by two.

    Given that 70 = 2 × 5 × 7 and 8 = 2 × 2 × 2, use a Venn diagram to find the lowest common multiple (LCM) of 70 and 8

Question

Use a Venn diagram to calculate the lowest common multiple (LCM) of 84 and 96

The calculations: Eighty four equals two multiplied by two multiplied by three multiplied by seven. Written below: Ninety six equals two multiplied by two multiplied by two multiplied by two multiplied by two multiplied by three.

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Practise the union of two sets

Practise using the union of two sets, with this quiz. You may need a pen and paper to help you with your answers.

Quiz

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