2.1 KiB
2.1 KiB
Page 515
Test Yourself
- If
Ris a relation fromAtoB,x \in A, andy \in B, the notationx R ymeans that ____.
x is related to y by R
- If
Ris a relation fromAtoB,x \in A, andy \in B, the notationx \cancel{R} ymeans that ____.
x is not related to y by R.
- If
Ris a relation fromAtoB,x \in A, andy \in B, the notation(y, x) \in R^{-1}if, and only if, ____.
(x, y) \in R - A relation on a set
Ais a relation from ____ to ____.
A; A
- If
Ris a relation on a setA, the directed graph ofRhas an arrow fromxtoyif, and only if, ____.
x is related to y by R
Page 526
Test Yourself
- For a relation
Ron a setAto be reflexive means that ____.
\forall x \in A, x R x
- For a relation
Ron a setAto be symmetric means that ____.
\forall x, y \in A, x R y \to y R x
- For a relation
Ron a setAto be transitive means that ____.
\forall x, y, z \in A, (x R y \wedge y R z) \to x R z
- To show that a relation
Ron an infinite setAis reflexive, you suppose that ____ and you show that ____.
x \in A; x R x
- To show that a relation
Ron an infinite setAis symmetric, you suppose that ____ and you show that ____.
\forall x, y \in A, x R y; y R x
- To show that a relation
Ron an infinite setAis transitive, you suppose that ____ and you show that ____.
\forall x, y, z \in A, x R y \wedge y R z; x R z
- To show that a relation
Ron a setAis not reflexive, you ____.
\exists x \in A, x \cancel{R} x
- To show that a relation
Ron a setAis not symmetric, you ____.
\exists x, y \in A, x R y \to y \cancel{R} x
- To show that a relation
Ron a setAis not transitive, you ____.
\exists x, y, z \in A, (x R y \wedge y R z) \to x \cancel{R} z
- Given a relation
Ron a setA, the transitive closure ofRis the relationR^tonAthat satisfies the following three properties: ____, ____, and ____.
R^t is transitive; R \subseteq R^t; if S is any other transitive relation
that contains R, then R^t \subseteq S