Verify that the inclusion relation $$ \subset $$ Is a partial ordering on the power set s where s={a,b,c}

To verify that the inclusion relation $ \subset $ is a partial ordering on the power set $ \mathcal{P}(S) $, where $ S = \{a, b, c\} $, we need to check if the relation satisfies the three properties of a partial order:

1. **Reflexivity**: For every subset $ A \in \mathcal{P}(S) $, $ A \subseteq A $ (every set is a subset of itself).
2. **Antisymmetry**: If $ A \subseteq B $ and $ B \subseteq A $, then $ A = B $.
3. **Transitivity**: If $ A \subseteq B $ and $ B \subseteq C $, then $ A \subseteq C $.

Let's verify these properties one by one.

### 1. Reflexivity
The power set $ \mathcal{P}(S) $ contains all the subsets of $ S $:
\[
\mathcal{P}(S) = \{\emptyset, \{a\}, \{b\}, \{c\}, \{a, b\}, \{a, c\}, \{b, c\}, \{a, b, c\}\}
\]
For each of these subsets $ A \in \mathcal{P}(S) $, it's clear that $ A \subseteq A $ holds (every set is a subset of itself). Hence, reflexivity is satisfied.

### 2. Antisymmetry
The antisymmetry property requires that if $ A \subseteq B $ and $ B \subseteq A $, then $ A = B $. In other words, if two sets are mutually subsets of each other, they must be equal.

For example:
- If $ A = \{a\} $ and $ B = \{a\} $, then clearly $ A = B $, since they are mutually subsets of each other.

In general, if $ A \subseteq B $ and $ B \subseteq A $, the only way this can happen is if the sets contain exactly the same elements. Thus, antisymmetry holds.

### 3. Transitivity
The transitivity property requires that if $ A \subseteq B $ and $ B \subseteq C $, then $ A \subseteq C $. This means that if $ A $ is a subset of $ B $ and $ B $ is a subset of $ C $, then $ A $ must also be a subset of $ C $.

For example:
- If $ A = \{a\}, B = \{a, b\}, C = \{a, b, c\} $, we can see that $ A \subseteq B $ and $ B \subseteq C $, and therefore $ A \subseteq C $.

This holds for all subsets in $ \mathcal{P}(S) $, so transitivity is satisfied.

### Conclusion
Since the inclusion relation $ \subset $ satisfies reflexivity, antisymmetry, and transitivity, it is indeed a **partial ordering** on the power set $ \mathcal{P}(S) $ of $ S = \{a, b, c\} $.

---

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. What is the difference between a partial order and a total order?
2. Can you give an example where a relation is reflexive and transitive but not antisymmetric?
3. How can we represent a partial order using a Hasse diagram?
4. If $ S $ had more elements, how would the power set and partial order relation change?
5. Can the inclusion relation $ \subset $ be a total order on some set?

**Tip:** A total order is a partial order where every pair of elements is comparable. In contrast, in a partial order, some elements might be incomparable.

Verify that the inclusion relation $$ \subset $$ is a partial ordering on the power set S where S = {a, b, c}

This problem verifies that the inclusion relation is a partial order on the power set of S = {a, b, c}. It covers set theory concepts such as reflexivity, antisymmetry, and transitivity, key properties of partial orders. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation