Question 1179822: Find all relations from {4, 5, 6} to {1, 2}.
Answer by CPhill(2264)   (Show Source):
You can put this solution on YOUR website! To find all relations from {4, 5, 6} to {1, 2}, we need to find all possible subsets of the Cartesian product of the two sets.
**1. Cartesian Product:**
The Cartesian product A × B of sets A = {4, 5, 6} and B = {1, 2} is:
A × B = {(4, 1), (4, 2), (5, 1), (5, 2), (6, 1), (6, 2)}
**2. Number of Relations:**
The number of relations from A to B is equal to the number of subsets of A × B. Since A × B has 6 elements, there are 2⁶ = 64 possible subsets.
**3. Listing the Relations:**
We can't list all 64 relations explicitly, but we can describe them:
* **Empty Relation:** {} (no pairs)
* **Relations with 1 pair:**
* {(4, 1)}
* {(4, 2)}
* {(5, 1)}
* {(5, 2)}
* {(6, 1)}
* {(6, 2)}
* **Relations with 2 pairs:**
* {(4, 1), (4, 2)}
* {(4, 1), (5, 1)}
* {(4, 1), (5, 2)}
* ... (and so on)
* **Relations with 3 pairs:**
* {(4, 1), (4, 2), (5, 1)}
* ... (and so on)
* ...
* **Full Relation:** {(4, 1), (4, 2), (5, 1), (5, 2), (6, 1), (6, 2)}
**General Representation:**
Any relation R from A to B is a subset of A × B. We can represent it as:
R ⊆ {(4, 1), (4, 2), (5, 1), (5, 2), (6, 1), (6, 2)}
To list them all, you would need to write out every possible combination of these pairs.
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