Which of these relations on {0*,* 1*,* 2*,* 3} are partial order-

ings? Determine the properties of a partial ordering that

the others lack.

**a)** {(0*,* 0)*,* (2*,* 2)*,* (3*,* 3)}

**b)** {(0*,* 0)*,* (1*,* 1)*,* (2*,* 0)*,* (2*,* 2)*,* (2*,* 3)*,* (3*,* 3)}

**c)** {(0*,* 0)*,* (1*,* 1)*,* (1*,* 2)*,* (2*,* 2)*,* (3*,* 1)*,* (3*,* 3)}

**d)** {(0*,* 0)*,* (1*,* 1)*,* (1*,* 2)*,* (1*,* 3)*,* (2*,* 0)*,* (2*,* 2)*,* (2*,* 3)*,*

(3*,* 0)*,* (3*,* 3)}

**e)** {(0*,* 0)*,* (0*,* 1)*,* (0*,* 2)*,* (0*,* 3)*,* (1*,* 0)*,* (1*,* 1)*,* (1*,* 2)*,*

(1*,* 3)*,* (2*,* 0)*,* (2*,* 2)*,* (3*,* 3)}

To determine whether each relation is a **partial ordering**, we need to check if the relation is:

1. **Reflexive**: Every element is related to itself, i.e., \((x, x)\) is in the relation for every \(x\) in the set.
2. **Antisymmetric**: If \((x, y)\) and \((y, x)\) are in the relation, then \(x = y\).
3. **Transitive**: If \((x, y)\) and \((y, z)\) are in the relation, then \((x, z)\) must also be in the relation.

The set in question is \(\{0, 1, 2, 3\}\). Let's evaluate each relation.

### **a)** \(\{(0, 0), (2, 2), (3, 3)\}\)

- **Reflexive**: Not reflexive because \( (1, 1) \) is missing.
- **Antisymmetric**: Yes, all pairs are of the form \((x, x)\), so antisymmetry holds trivially.
- **Transitive**: Yes, since there are no pairs like \((x, y)\) and \((y, z)\) for \(x \neq y\), transitivity holds trivially.
  
**Conclusion**: This is **not a partial ordering** because it is not reflexive.

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### **b)** \(\{(0, 0), (1, 1), (2, 0), (2, 2), (2, 3), (3, 3)\}\)

- **Reflexive**: Not reflexive because \((3, 3)\) is present but \((1, 1)\) is missing.
- **Antisymmetric**: Yes, antisymmetry holds trivially_ans_parser

Which of these relations on {0, 1, 2, 3} are partial orderings? Determine the properties of a partial ordering that the others lack. a) {(0, 0), (2, 2), (3, 3)} b) {(0, 0), (1, 1), (2, 0), (2, 2), (2, 3), (3, 3)} c) {(0, 0), (1, 1), (1, 2), (2, 2), (3, 1), (3, 3)} d) {(0, 0), (1, 1), (1, 2), (1, 3), (2, 0), (2, 2), (2, 3), (3, 0), (3, 3)} e) {(0, 0), (0, 1), (0, 2), (0, 3), (1, 0), (1, 1), (1, 2), (1, 3), (2, 0), (2, 2), (3, 3)}

Explore the process of determining whether relations on the set {0, 1, 2, 3} form partial orderings by evaluating their reflexive, antisymmetric, and transitive properties. This analysis is crucial in understanding key concepts in discrete mathematics, suitable for undergraduate students.

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