1. What Are Equivalent Sets?
Equivalent sets are sets that have the same number of elements, even if the actual elements are different. The elements do not need to match — only the count matters.
The symbol for equivalence is usually \( \sim \). If set A and set B have the same number of elements, we write:
\( A \sim B \)
2. Key Idea Behind Equivalent Sets
Two sets are equivalent when their cardinalities are equal. In simple terms:
\( A \sim B \iff |A| = |B| \)
This makes it easy to compare sets without looking at the actual elements.
3. Examples of Equivalent Sets
Here are some sets that are equivalent because they contain the same number of elements:
3.1. Basic Examples
- → Both have 3 elements
\( \{1, 2, 3\} \sim \{a, b, c\} \)
- → Both have 2 elements
\( \{10, 20\} \sim \{x, y\} \)
- \{red, blue, green, yellow\} \sim \{2, 4, 6, 8\}
4. Non-Examples (Not Equivalent)
Two sets are not equivalent when the number of elements is different.
4.1. Examples
- → 3 elements vs 2
\( \{1,2,3\} \not\sim \{7,8\} \)
- → 2 elements vs 4
\( \{a,b\} \not\sim \{1,2,3,4\} \)
5. How to Check if Two Sets Are Equivalent
To check equivalence, compare only the number of elements. The actual values do not matter.
5.1. Steps
- Count the number of elements in the first set.
- Count the number of elements in the second set.
- If the counts are the same, the sets are equivalent.
5.2. Example
Are \( \{2,4,6,8\} \) and \( \{a,b,c,d\} \) equivalent?
Both have 4 elements → Yes, they are equivalent.
6. Important Notes
A few useful points about equivalent sets:
6.1. Key Ideas
- The elements do not need to be the same — only the count matters.
- Equal sets are always equivalent, but equivalent sets are not always equal.
- Equivalent sets have the same cardinality.
- Works for both finite and infinite sets, though infinite equivalence is studied at higher levels.