# 4. Task and order of execution / Завдання та порядок виконання

> **Source of tasks.** The seven tasks below are taken from the assignment; Tasks 6–7
> extend it with antisymmetry and partial orders. Implement the described behaviour —
> the input/output examples are authoritative. Task numbers follow the assignment, so
> within a difficulty band they are not always consecutive.

## Order of execution

The tasks are grouped into three difficulty bands. Implement them in order and
test each against its example; your score corresponds to the band of tasks you
complete correctly.

## Easy tasks (60–74 points)

### Task 1 — Reflexivity Checker
**Objective:** Write a program that checks if a relation is reflexive on a set.
- **Input:** Set: `{1, 2, 3}`; Relation: `{(1, 1), (2, 2), (3, 3)}`
- **Output:** Is Reflexive: `True`

### Task 2 — Symmetry Identifier
**Objective:** Implement a program that identifies if a relation on a set is
symmetric.
- **Input:** Relation: `{(1, 2), (2, 1), (3, 3)}`
- **Output:** Is Symmetric: `True`

### Task 3 — Transitivity Verifier
**Objective:** Create a program to verify the transitivity of a relation on a set.
- **Input:** Relation: `{(1, 2), (2, 3), (1, 3)}`
- **Output:** Is Transitive: `True`

## Medium tasks (75–89 points)

### Task 4 — Equivalence Relation Checker
**Objective:** Create a program that checks if a relation on a set is an
equivalence relation (reflexive, symmetric, and transitive).
- **Input:** Set: `{1, 2, 3}`; Relation: `{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2), (1, 3), (3, 1)}`
- **Output:** Is Equivalence Relation: `True`

### Task 6 — Antisymmetry Checker
**Objective:** Implement a program that checks whether a relation on a set is
antisymmetric (no two **distinct** elements are related both ways).
- **Input:** Relation: `{(1, 1), (1, 2), (2, 3)}`
- **Output:** Is Antisymmetric: `True`

## Hard tasks (90–100 points)

### Task 5 — Inverse Relation Generator
**Objective:** Implement a program that generates the inverse of a given relation.
- **Input:** Relation: `{(1, 2), (3, 4), (5, 6)}`
- **Output:** Inverse Relation: `{(2, 1), (4, 3), (6, 5)}`

### Task 7 — Partial Order Checker
**Objective:** Check whether a relation on a set is a **partial order** (reflexive,
antisymmetric, and transitive).
- **Input:** Set: `{1, 2, 3}`; Relation: `{(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}`
- **Output:** Is Partial Order: `True`