1-Deterimine which of the following are equivalence relations and/or partia orde
ID: 2963343 • Letter: 1
Question
1-Deterimine which of the following are equivalence relations and/or partia ordering relations for the given sets:
a- A={Lines in the plane}; x r y if and only if x is parallel to y.
b- A=R; if and only if x is parallel to y. (where R is real numbers)
2- For the set of cities on a map, consider the relation x r y if and only if city x is connected by a city y. A city is considered to be connected to itself, and two cities are connected even though there are cities on the road between them. Is this an equivalence relation or a partial ordering? Explain?
Explanation / Answer
1
a) A is an Equivalence Relation
Reflexive : x is parallel to itself => x R x
Symmetric : x is parallel to y => y is parallel to x.
Therefore x R y => y R x
Transitive : x is parallel to y and y is parallel to z then x, y, z are parallel to each other.
=> x R y and y R z => x R z
Therefore A is equivalent.
2
It is an Equivalence Relation.
Reflexive : x is connected to itself (Given) => x R x
Symmetric : x is connected to y => y is connected to x as two cities are inter-connected.
=> x R y => y R x
Transitive : As it is given that two cities are connected even if there are cities in between, we say that if x is connected to y and y is connected to z then x is connected to z.
=> x R y and y R z => x R z
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