MAT2612
ASSIGNMENT 04
Unique No:872328
Due 2025
, Discrete Mathematics Assignment
Question 1: Determine whether the relation R is a par-
tial order on the set A, where A = R and aRb if and only
if a ≤ b.
Problem Statement
Determine whether the relation R on the set A = R, defined by aRb if and only if a ≤ b, is
a partial order.
Step 1: Understand the definition of a partial order
A relation R on a set A is a partial order if it satisfies three properties:
• Reflexivity: For all a ∈ A, aRa (every element is related to itself).
• Antisymmetry: If aRb and bRa, then a = b for all a, b ∈ A.
• Transitivity: If aRb and bRc, then aRc for all a, b, c ∈ A.
We need to check if the relation R, defined by a ≤ b, satisfies these properties on A = R.
Step 2: Check reflexivity
For R to be reflexive, every real number a ∈ R must satisfy aRa, i.e., a ≤ a.
• For any real number a, it is true that a ≤ a (e.g., 1 ≤ 1, −2 ≤ −2, 0 ≤ 0).
• This holds for all a ∈ R, so R is reflexive.
Step 3: Check antisymmetry
For R to be antisymmetric, if aRb and bRa, i.e., a ≤ b and b ≤ a, then a = b.
• If a ≤ b and b ≤ a, then by the properties of real numbers, a = b (e.g., if 2 ≤ 3 and
3 ≤ 2, then 2 = 3, which is false unless a = b).
• This holds for all a, b ∈ R, so R is antisymmetric.
1
, Step 4: Check transitivity
For R to be transitive, if aRb and bRc, i.e., a ≤ b and b ≤ c, then aRc, i.e., a ≤ c.
• If a ≤ b and b ≤ c, then by the transitive property of the real numbers, a ≤ c (e.g., if
1 ≤ 2 and 2 ≤ 3, then 1 ≤ 3).
• This holds for all a, b, c ∈ R, so R is transitive.
Step 5: Conclusion
Since the relation R satisfies reflexivity, antisymmetry, and transitivity, it is a partial order
on the set A = R.
Final Answer
The relation R, defined by aRb if and only if a ≤ b, is a partial order on the set A = R.
2
ASSIGNMENT 04
Unique No:872328
Due 2025
, Discrete Mathematics Assignment
Question 1: Determine whether the relation R is a par-
tial order on the set A, where A = R and aRb if and only
if a ≤ b.
Problem Statement
Determine whether the relation R on the set A = R, defined by aRb if and only if a ≤ b, is
a partial order.
Step 1: Understand the definition of a partial order
A relation R on a set A is a partial order if it satisfies three properties:
• Reflexivity: For all a ∈ A, aRa (every element is related to itself).
• Antisymmetry: If aRb and bRa, then a = b for all a, b ∈ A.
• Transitivity: If aRb and bRc, then aRc for all a, b, c ∈ A.
We need to check if the relation R, defined by a ≤ b, satisfies these properties on A = R.
Step 2: Check reflexivity
For R to be reflexive, every real number a ∈ R must satisfy aRa, i.e., a ≤ a.
• For any real number a, it is true that a ≤ a (e.g., 1 ≤ 1, −2 ≤ −2, 0 ≤ 0).
• This holds for all a ∈ R, so R is reflexive.
Step 3: Check antisymmetry
For R to be antisymmetric, if aRb and bRa, i.e., a ≤ b and b ≤ a, then a = b.
• If a ≤ b and b ≤ a, then by the properties of real numbers, a = b (e.g., if 2 ≤ 3 and
3 ≤ 2, then 2 = 3, which is false unless a = b).
• This holds for all a, b ∈ R, so R is antisymmetric.
1
, Step 4: Check transitivity
For R to be transitive, if aRb and bRc, i.e., a ≤ b and b ≤ c, then aRc, i.e., a ≤ c.
• If a ≤ b and b ≤ c, then by the transitive property of the real numbers, a ≤ c (e.g., if
1 ≤ 2 and 2 ≤ 3, then 1 ≤ 3).
• This holds for all a, b, c ∈ R, so R is transitive.
Step 5: Conclusion
Since the relation R satisfies reflexivity, antisymmetry, and transitivity, it is a partial order
on the set A = R.
Final Answer
The relation R, defined by aRb if and only if a ≤ b, is a partial order on the set A = R.
2
