status: DONE
created: 2024-01-16T21:03
updated: 2024-06-11T01:14
discrete-mathematics-assignment-7
Determine whether the following relation R is reflexive, symmetric, antisymmetric, or transitive if (remember to prove your answer)
a)
The following relation R is reflexive, symmetric, anti-symmetric, and transitive
R is reflexive: False, since there are no ordered pairs in R.
R is symmetric: True, vacuously. There are no ordered pairs in R to violate symmetry.
R is antisymmetric: True, vacuously. There are no ordered pairs in R to violate antisymmetry.
R is transitive: True, vacuously. There are no ordered pairs in R to violate transitivity.
b)
The following relation R is symmetric.
c)
The following relation R is reflexive, symmetric and transitive
then
d)
The following relation R is reflexive, symmetric and transitive
And
e)
The following relation R is reflexive and symmetric
Let
a) if
A relation 𝑅 on a set 𝐴 is called reflexive if (𝑎, 𝑎) ∈ 𝑅 for every element 𝑎 ∈ 𝐴.
b) if
A relation 𝑅 on a set 𝐴 is called symmetric if (𝑏, 𝑎) ∈ 𝑅 whenever (𝑎, 𝑏) ∈ 𝑅.
Otherwise,
c) if
A relation 𝑅 on a set 𝐴 is called anti-symmetric if whenever (𝑎, 𝑏) ∈ 𝑅 and (𝑏, 𝑎) ∈ 𝑅, then 𝑎 = 𝑏
Otherwise,
d) if
A relation 𝑅 on a set 𝐴 is called transitive if whenever (𝑎, 𝑏) ∈ 𝑅 and (𝑏, 𝑐) ∈ 𝑅, then (𝑎, 𝑐) ∈ 𝑅.
Determine whether the following relations are equivalence relation. You need to prove your answer. If yes, list all distinct equivalent classes (each subset of the partition defined by the equivalence relation).
if (x mod 5=x)
if (x mod 5=y) y mod 5 may not be x
For reflexive, the same student must be in the same faculty/school as himself/herself, so R satisfies reflexivity
For symmetric, if student a and student b are from the same faculty/school, then student b and student a are from the same faculty/school, then (a, b) ∈ R, (b, a) ∈ R, so R satisfies symmetric.
For transitive, if (a, b) ∈ R, (b, c) ∈ R, then will have (a, c) ∈ R, so R satisfies transitive.
∴ this relation is equivalence relation.
For reflexive, let A = B, that will have a bijection
For symmetric, if
(A, B) ∈ R, that will also have (B, A) ∈ R, so R satisfies symmetric
For transitive, if
ℎ: A → C is a bijection, so R satisfies transitive.
∴ this relation is equivalence relation. equivalent classes are all