Question

The subgroup relation ≤ on the set of subgroups G is reflexive, transitive, and anti-symmetric.

Answer #1

Let H be a subgroup of G. Now, every group is a subgroup of itself. Hence, . Hence the subgroup relation is reflexive..

Now, let H_{1}, H_{2}, H_{3} be three
subgroups of G such that
. Now, since H_{1} is a subgroup of H_{2}, and
H_{2} is a subgroup of H_{3}, therefore,
H_{1} must be a subgroup of H_{3}. That is,
. Therefore, the subgroup relation is
**transitive.**

Now, let
. That is, H_{1} must also be a subset of H_{2} and
H_{2} must be a subset of H_{1}. This is only
possible if H_{1}=H_{2}. Hence, the subgroup
relation is **anti-symmetric.**

show that the relation "≈" is reflexive, symmetric, and
transitive on the class of all sets.

Disprove: The following relation R on set Q is either reflexive,
symmetric, or transitive. Let t and z be elements of Q. then t R z
if and only if t = (z+1) * n for some integer n.

Determine whether the relation R is reflexive, symmetric,
antisymmetric, and/or transitive [4 Marks]
22
The relation R on Z where (?, ?) ∈ ? if ? = ? .
The relation R on the set of all subsets of {1, 2, 3, 4} where
SRT means S C T.

Consider the relation R= {(1,2),(2,2),(2,3),(3,1),(3,3)}. Is R
transitive, not reflexive, symmetric or equivalence relation?

4. What is a group action? Which subgroups of a symmetric group
are called transitive? What is an example of a transitive subgroup
of S6 that is not S6 itself?

Prove that strong connectivity is reflexive, transitive and
symmetric.

Show that reflexive sentences are independent from symmetric,
and transitive sentences by constructing a structure that satisfy
symmetric and transitive quality but not reflexive.
Reflexive :∀xE(x, x)
symmetric :∀xy(E(x, y) → E(y, x))
transitive: ∀xyz(E(x, y) ∧ E(y, z) → E(x, z))

Show that symmetric sentences are independent from reflexive,
and transitive sentences by constructing a structure that satisfy
transitive and reflexive quality but not symmetric.
Reflexive :∀xE(x, x)
symmetric :∀xy(E(x, y) → E(y, x))
transitive: ∀xyz(E(x, y) ∧ E(y, z) → E(x, z))

Prove or disprove: The relation "is-a-normal-subgroup-of" is a
transitive relation.

For each of the properties reflexive, symmetric, antisymmetric,
and transitive, carry out the following.
Assume that R and S are nonempty relations on a set A that both
have the property. For each of Rc, R∪S, R∩S, and R−1, determine
whether the new relation
must also have that property;
might have that property, but might not; or
cannot have that property.
A ny time you answer Statement i or Statement iii, outline a
proof. Any time you answer Statement ii,...

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