1. What Is the Difference of Sets?
The difference of two sets shows the elements that belong to the first set but not to the second set. It removes anything that the two sets share.
The difference is not the same as subtraction, but it follows a similar idea — taking away common elements.
2. Notation
The symbol for the difference of sets is:
\( A - B \)
This is read as “A minus B”.
3. Meaning of A − B
\( A - B \) contains:
- Elements in A
- Elements not in B
Any element found in both sets is removed.
4. Examples
Here are some simple examples:
4.1. Example 1
\( A = \{1,2,3\}, B = \{3,4\} \Rightarrow A - B = \{1,2\} \)
4.2. Example 2
\( X = \{a,b,c\}, Y = \{b,d\} \Rightarrow X - Y = \{a,c\} \)
4.3. Example 3 (Empty Difference)
If all elements of A appear in B:
\( P = \{2,4\}, Q = \{2,4,6\} \Rightarrow P - Q = \emptyset \)
5. Difference Works One Way
The difference of sets is not symmetric. This means:
\( A - B \neq B - A \)
because the order matters.
5.1. Example
Using:
A = \{1,2,3\}, B = \{3,4\}
Then:
- A − B = {1,2}
- B − A = {4}
6. Difference with More Than Two Sets
We can also take the difference across multiple sets.
6.1. Example
A = \{1,2,3,4\}, B = \{2\}, C = \{4\}
Then:
\( A - (B \cup C) = \{1,3\} \)
7. Difference in Venn Diagrams
In a Venn diagram, the difference A − B is shown by shading only the part of A that does not overlap with B.
8. Important Points
A few things to remember about the difference of sets:
8.1. Key Ideas
- A − B contains elements in A but not in B.
- If A and B do not overlap, then A − B is just A.
- The difference A − B is not the same as B − A.
- Difference helps in filtering out unwanted elements.