~/writing/math-foundations-for-reasoning-models $ ls
Sets, Relations, and Functions
Equality
Set equality means two sets contain exactly the same elements.
~/writing/math-foundations-for-reasoning-models $ cat topic.md
Equality
Definition
Two sets are equal when they contain exactly the same elements.
Example:
A = {1, 2, 3}
B = {3, 2, 1}
These sets are equal.
The order is different, but the elements are the same.
Set equality = same elements.
Order Does Not Affect Equality
These are equal:
{a, b, c}
{c, b, a}
Why?
Because every element in the first set is in the second set, and every element in the second set is in the first set.
Sets are equal by membership, not written order.
Repetition Does Not Affect Equality
These describe the same set:
{1, 1, 2, 3}
{1, 2, 3}
The repeated 1 does not create a new element.
A set either contains 1 or it does not.
Repetition does not change set equality.
Equality Through Subsets
Two sets are equal if each is a subset of the other.
Example:
A = {1, 2, 3}
B = {3, 2, 1}
A is a subset of B.
B is a subset of A.
Therefore:
A = B
Mutual subset relation proves set equality.
Final Summary
- Two sets are equal when they contain exactly the same elements.
- Order does not affect set equality.
- Repetition does not affect set equality.
- Set equality is based on membership.
- Two sets are equal when each is a subset of the other.
~/writing/math-foundations-for-reasoning-models $ open
Continue the series.
Power Set