# Difference of sets

This lesson will explain how to find the difference of sets. We will start with a definition

Definition:

Given set A and set B the set difference of set B from set A is the set of all element in A, but not in B.

We can write A − B

Example #1.

Take a close look at the figure above. Elements in A only are b, d, e,g.

Therefore, A − B = { b, d, e, g}

Notice that although elements a, f, c are in A, we did not include them in A − B because we must not take anything in set B.

Sometimes, instead of looking at a the Venn Diagrams, it may be easier to write down the elements of both sets

Then, we show in bold the elements that are in A, but not in B

A = {b, d, e, g, a, f, c}

B = { k, h, u, a, f, c}

Example #2.

Find B − A

Notice that this time you are looking for anything you see in B only

Elements that are in B only are shown in bold below

Let A = {1 orange, 1 pinapple, 1 banana, 1 apple}

Let B = {1 orange, 1 apricot, 1 pinapple, 1 banana, 1 mango, 1 apple, 1 kiwifruit }

B − A = {1 apricot, 1 mango, 1 kiwifruit}

Example #3.

Find A − B

B = { 1, 2, 4, 6}

A = {1, 2, 4, 6, 7, 8, 9 }

What I see in A that are not in B are 7, 8, and 9

A − B = { 7, 8, 9}

Example #4.

Find B − A

A = { x / x is a number bigger than 6 and smaller than 10}

B = { x / x is a positive number smaller than 15}

A = {7, 8, 9} and B = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}

Everything you see in bold above are in B only.

B − A = {1, 2, 3, 4, 5, 6, 10, 11, 12, 13, 14}

The graph below shows the shaded region for A − B and B − A

This ends the lesson about the difference of sets.

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