Sets Intersection: Intersection Of Three Sets

The intersection of three sets X, Y and Z is the set of elements that are common to sets X, Y and Z. It is denoted by X ∩ Y ∩ Z

Example:

Draw a Venn diagram to represent the relationship between the sets

X = {1, 2, 5, 6, 7, 9}, Y = {1, 3, 4, 5, 6, 8} and

Z = {3, 5, 6, 7, 8, 10}

Solution:

We find that X ∩ Y ∩ Z = {5, 6}, X ∩ Y = {1, 5, 6},

Y ∩ Z = {3, 5, 6, 8} and X ∩ Z = {5, 6, 7}

For the Venn diagram:

Step 1 : Draw three overlapping circles to represent the three sets.

Step 2 : Write down the elements in the intersection X ∩ Y ∩ Z

Step 3 : Write down the remaining elements in the intersections:

X ∩ Y, Y ∩ Z and X ∩ Z

Step 4 : Write down the remaining elements in the respective sets.

Again, notice that you start filling the Venn diagram from the elements in the intersection first.

In general, there are many ways that 3 sets may intersect. Some examples are shown below.

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