Consider the set of all people, and the following relation RR on this set:
⟨x,y⟩∈R⟨x,y⟩∈R if yy is the father of xx
Pick the right properties of RR:

Reflexive

Symmetric

Transitive

Total

Irreflexive:   ∀x : ⟨x,x⟩∉R  ∀x : ⟨x,x⟩∉R

Asymmetric:   ∀x,y : ⟨x,y⟩∈R → ⟨y,x⟩∉R  ∀x,y : ⟨x,y⟩∈R → ⟨y,x⟩∉R

Anti-symmetric

Intransitive:   ∀x,y,z : (⟨x,y⟩∈R ∧ ⟨y,z⟩∈R) → ⟨x,z⟩∉R


Let's examine the relation \( R \) where \( \langle x, y \rangle \in R \) means that \( y \) is the father of \( x \). We need to check which of the listed properties apply to this relation.

### Reflexive
- A relation is **reflexive** if \( \langle x, x \rangle \in R \) for all \( x \). In this case, this would mean that everyone is their own father, which is clearly not true.

**Conclusion**: The relation is **not reflexive**.

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### Symmetric
- A relation is **symmetric** if \( \langle x, y \rangle \in R \) implies \( \langle y, x \rangle \in R \). In this case, if \( y \) is the father of \( x \), it does not imply that \( x \) is the father of \( y \). Hence, the relation is not symmetric.

**Conclusion**: The relation is **not symmetric**.

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### Transitive
- A relation is **transitive** if \( \langle

Consider the set of all people, and the following relation R on this set: ⟨x, y⟩ ∈ R if y is the father of x. Pick the right properties of R: Reflexive, Symmetric, Transitive, Total, Irreflexive, Asymmetric, Anti-symmetric, Intransitive.

Explore the properties of the father-child relation in set theory. This problem investigates whether the relation is reflexive, symmetric, transitive, irreflexive, asymmetric, and more. Suitable for advanced high school or undergraduate students studying set theory and relations.

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Solution

Reflexive

Symmetric

Transitive

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