Let $S$ be any non-empty set and $R$ be a relation defined on the power set of $S$, i.e., on $P(S)$ by $A\,R\,B$ iff $A \subset B$ for all $A, B \in P(S)$. Show that $R$ is reflexive and transitive but not symmetric.

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> [!TLDR]
> The subset relation on the power set $P(S)$ is reflexive and transitive but fails symmetry, shown with an explicit counterexample.

> [!FORMULA]
> On $P(S)$ (the power set of $S$), define $A\,R\,B \iff A\subseteq B$.
> - **Reflexive:** $(A,A)\in R\ \forall A\in P(S)$
> - **Symmetric:** $(A,B)\in R\Rightarrow(B,A)\in R$
> - **Transitive:** $(A,B)\in R,(B,C)\in R\Rightarrow(A,C)\in R$

1. **Check Reflexivity.** For any $A\in P(S)$, every element of $A$ is (trivially) an element of $A$, so $A\subseteq A$ by definition of subset. Hence
 $$A\,R\,A\quad	ext{for every } A\in P(S)$$
 So $R$ **is reflexive**.

2. **Check Transitivity.** Suppose $A\,R\,B$ and $B\,R\,C$, i.e. $A\subseteq B$ and $B\subseteq C$. Take any element $x\in A$. Since $A\subseteq B$, $x\in B$. Since $B\subseteq C$, $x\in C$. So every $x\in A$ is also in $C$:
 $$A\subseteq C\ \Rightarrow\ A\,R\,C$$
 So $R$ **is transitive**.

3. **Check (non-)Symmetry with a counterexample.** Let $S=\{1,2\}$, so $P(S)=\{\varnothing,\{1\},\{2\},\{1,2\}\}$. Take $A=\varnothing$ and $B=\{1\}$.
 - $A\subseteq B$: the empty set is a subset of every set, so $\varnothing\subseteq\{1\}$ is **true**, i.e. $A\,R\,B$ holds.
 - $B\subseteq A$: is $\{1\}\subseteq\varnothing$? No — $1\in\{1\}$ but $1
otin\varnothing$, so this is **false**, i.e. $B\,R\,A$ does **not** hold. …

Type	Reflexive?	Symmetric?	Transitive?	Example
Empty relation	No (unless $A = \emptyset$ )	Yes (vacuously)	Yes (vacuously)	$R = \emptyset$ on $A = \{1,2\}$
Universal relation	Yes	Yes	Yes	$R = A \times A$
Identity relation	Yes	Yes	Yes	$R = \{(a,a) \mid a \in A\}$
Equivalence relation	Yes	Yes	Yes	"Same city", "same remainder mod 3"

Q.Let $S$ be any non-empty set and $R$ be a relation defined on the power set of $S$ , i.e., on $P(S)$ by $A\,R\,B$ iff $A \subset B$ for all $A, B \in P(S)$ . Show that $R$ is reflexive and transitive but not symmetric.

Concept understanding — Types of Relations

Types of Relations – From Intuition to Precision

The Intuition: Three Key Properties

The Precise Definitions

The Four Types of Relations You Must Know

Equivalence Relations — The Most Important Type

Step 1 — Recall the two definitions

Step 2 — Test option (A)

Step 3 — Eliminate the others

Step 1 — Match the statement to a definition

Step 2 — Rule out the other properties

Step 3 — Conclude …

Q.Let SSS be any non-empty set and RRR be a relation defined on the power set of SSS, i.e., on P(S)P(S)P(S) by A R BA\,R\,BARB iff A⊂BA \subset BA⊂B for all A,B∈P(S)A, B \in P(S)A,B∈P(S). Show that RRR is reflexive and transitive but not symmetric.

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Q.Let $S$ be any non-empty set and $R$ be a relation defined on the power set of $S$ , i.e., on $P(S)$ by $A\,R\,B$ iff $A \subset B$ for all $A, B \in P(S)$ . Show that $R$ is reflexive and transitive but not symmetric.

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