Using Venn diagrams
A Venn diagram shows the relationship between different sets or categories of data.
For example, the following list of numbers can be sorted depending on whether the numbers are even or not, and whether or not they are multiples of 3.
1, 3, 4, 5, 6, 8, 9, 14, 18
The numbers 1 and 5 are neither even nor multiples of 3, so they are placed outside the rings of the Venn diagram. The numbers 6 and 18 are multiples of 3 that are also even numbers, so they are placed in the overlap (or intersection) of the two circles.
Venn diagrams can be used to calculate the The highest common factor (HCF) of two numbers is the largest number which will divide exactly into both of them, for example, the highest common factor of 24 and 36 is 12. and The smallest positive number that is a multiple of two or more numbers. of numbers.
Example
Find the HCF and LCM of 24 and 180.
Break the numbers into the To multiply. The product of two numbers is the answer to the multiplication of the numbers. The product of 5 and 8 is 40. of prime factors using prime factor trees.
The product of prime factors for 24 is: \(2 \times 2 \times 2 \times 3\)
The product of prime factors for 180 is: \(2 \times 2 \times 3 \times 3 \times 5\)
Put each prime factor in the correct place in the Venn diagram. Any common factors should be placed in the intersection of the two circles.
The highest common factor is found by multiplying together the numbers in the intersection of the two circles.
HCF = \(2 \times 2 \times 3 = 12\)
The LCM is found by multiplying together the numbers from all three sections of the circles.
LCM = \(2 \times 2 \times 2 \times 3 \times 3 \times 5 = 360\)
The circles of a Venn diagram do not necessarily contain all the outcomes. Drawing a rectangular frame around the circles enables those outcomes to be recorded outside of the circles but inside the rectangle. The universal set symbol \( \xi \) shows that the rectangle includes all the possible outcomes.