1. What Are Equal Sets?
Equal sets are sets that contain exactly the same elements. It does not matter how the elements are arranged or written. If every element of one set is also present in the other set, and vice versa, the two sets are equal.
The idea is simple: sets ignore order and repetitions, so only the elements matter.
1.1. Key Condition
Two sets A and B are equal if:
\( A = B \iff (\forall x, x \in A \Rightarrow x \in B) \text{ and } (x \in B \Rightarrow x \in A) \)
2. Examples of Equal Sets
Here are simple examples where two sets have the same elements, even if written differently.
2.1. Basic Examples
\( A = \{1, 2, 3\}, B = \{3, 2, 1\} \Rightarrow A = B \)
\( X = \{a, b, c\}, Y = \{b, c, a\} \Rightarrow X = Y \)
- \{red, blue, green\} = \{green, red, blue\}
2.2. Examples With Repetition
Repetition does not change a set.
\( \{1,1,2,2,3\} = \{1,2,3\} \)
\( \{a,a,a,b\} = \{a,b\} \)
3. Non-Examples (Not Equal)
Sometimes two sets look similar but are not equal because one contains an extra element or is missing one.
3.1. Examples
- → B has one extra element
\( A = \{1,2,3\}, B = \{1,2,3,4\} \)
- → D is missing an element
\( C = \{a,b,c\}, D = \{a,c\} \)
4. How to Check if Two Sets Are Equal
You can check equality by comparing elements directly:
4.1. Steps
- Check that every element of the first set appears in the second set.
- Check that every element of the second appears in the first.
- If both conditions are true, the sets are equal.
4.2. Example
Check if \( \{2,4,6\} \) and \( \{6,4,2\} \) are equal.
Both contain 2, 4, and 6. No element is different → They are equal.
5. Important Notes
A few useful points to remember:
5.1. Key Ideas
- Order does not matter in sets.
- Repetition does not matter.
- Two sets may look different at first but still be equal.
- Equal sets always have the same cardinality.