For each relation below, explain why the relation does or does not satisfy each
ID: 3693337 • Letter: F
Question
For each relation below, explain why the relation does or does not satisfy each of the properties reflexive, symmetric, antisymmetric, and transitive.
(a) “isBrotherOf” on the set of people.
(b) “isFatherOf” on the set of people.
(c) The relation R = {hx, yi | x 2 + y 2 = 1} for real numbers x and y.
(d) The relation R = {hx, yi | x 2 = y 2} for real numbers x and y.
(e) The relation R = {hx, yi | x mod y = 0} for x, y {1, 2, 3, 4}.
(f) The empty relation (i.e., the relation with no ordered pairs for which it is true) on the set of integers.
(g) The empty relation (i.e., the relation with no ordered pairs for which it is true) on the empty set.
Explanation / Answer
Answer:
a)
Reflexive : no person can be brother of himself.
Symmetric: no female can be brother of another female.
Antisymmetric: if a is the brother of b then b can also be the brother of a util unless a is female.
Transitive : it is transitive
Since a,b nd b,c imply a,c
a,b b,c possible only if all are males.
b)
No person can be father of himself hence not reflexive
Not symmetric as no female can be father of another female.
Antisymmetric : yes it is as only one property can hold as out of a,b onky one can be father of another or none
But not both.
Transitive: it cant be as if a is the father of b and b is the father of c then a is the grandfather of c not father.
c)
The relation is not reflexive since x2+x2 =1implies x is not real.
It is symmetric since if x2+y2=1 then y2+x2=1
It is not antisymmetric since it is symmetric.
It is not transitive since x2+y2=1 and y2+z2=1 not necessarily imples x2+z2=1.
d)
It is reflexive since
X2=x2
It is symmetric also since if x2=y2 then y2=x2
It is notantisymmetric since it is symmetric
It is transitive since if x2=y2 and y2=z2 then x2=z2
e)
it is reflexive since xmod x=0
It is not symmetric since if xmody=0 then ymodx cant be 0 .
It is antisymmetric since out of xmody or ymodx only one can be 0 or none.
It is transitive also since 4mod2=0, 2mod1=0 hence 4mod1=0.
For f) and g) , check this :
Whether the empty relation is reflexive or not depends on the set on which you are defining this relation -- you can define the empty relation on any set XX.
The statement "RR is reflexive" says: for each xXxX, we have (x,x)R(x,x)R. This is vacuously true if X=X=, and it is false if XX is nonempty.
The statement "RR is symmetric" says: if (x,y)R(x,y)R then (y,x)R(y,x)R. This is vacuously true, since (x,y)R(x,y)R for all x,yXx,yX.
The statement "RR is transitive" says: if (x,y)R(x,y)R and (y,z)R(y,z)R then (x,z)R(x,z)R. Similarly to the above, this is vacuously true.
To summarize, RR is an equivalence relation if and only if it is defined on the empty set. It fails to be reflexive if it is defined on a nonempty set.
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