Question

Problem 10.2. Let R be a relation on the set X (so $R \subseteq X \times X$). a) Show that the following are equivalent: (i) The relation R is both symmetric and anti-symmetric. (ii) We have $R \subseteq \Delta_X := \{(x, x) \mid x \in X\}$. b) Show that the following are equivalent: (i) The relation R is both an equivalence relation and a partial ordering. (ii) We have $R = \Delta_X$. c) Show: if X has at least two elements, there is no relation on R that is both an equivalence relation and a total ordering.

          Problem 10.2. Let R be a relation on the set X (so $R \subseteq X \times X$).
a) Show that the following are equivalent:
(i) The relation R is both symmetric and anti-symmetric.
(ii) We have $R \subseteq \Delta_X := \{(x, x) \mid x \in X\}$.
b) Show that the following are equivalent:
(i) The relation R is both an equivalence relation and a partial ordering.
(ii) We have $R = \Delta_X$.
c) Show: if X has at least two elements, there is no relation on R that is both
an equivalence relation and a total ordering.

Problem 10.2. Let R be a relation on the set X (so R ⊆ X × X).
a) Show that the following are equivalent:
(i) The relation R is both symmetric and anti-symmetric.
(ii) We have R ⊆ := {(x, x) | x ∈ X}.
b) Show that the following are equivalent:
(i) The relation R is both an equivalence relation and a partial ordering.
(ii) We have R =.
c) Show: if X has at least two elements, there is no relation on R that is both
an equivalence relation and a total ordering.

Added by Rita W.

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