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 $S = \{a, b, c\}$.

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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.