Selected Exercises Copyright Peter Cappello 2011 2 Exercise 10 Which relations in Exercise 4 are irreflexive A relation is irreflexive a A a a R Ex 4 relations on the set of all people ID: 525899
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Slide1
Relations & Their Properties: Selected ExercisesSlide2
Copyright © Peter Cappello 2011
2
Exercise 10
Which relations in Exercise 4 are irreflexive?
A relation is
irreflexive
a
A (a, a)
R.
Ex. 4 relations on the set of all people:
a
is taller than
b
.
a
and
b
were born on the same day.
a
has the same first name as
b
.
a
and
b
have a common grandparent.Slide3
Copyright © Peter Cappello 2011
3
Exercise 20
Must an asymmetric relation be antisymmetric?
A relation is
asymmetric
a
b ( aRb
(b, a)
R ).Slide4
Copyright © Peter Cappello 2011
4
Exercise 20
Must an asymmetric relation be antisymmetric?
A relation is
asymmetric
a
b ( aRb
(b, a)
R ).
To Prove:
(
a
b ( aRb
(b, a)
R ) )
(
a
b ( (aRb
bRa )
a = b ) )
Proof:
Assume R is asymmetric.
a
b ( ( a, b )
R
( b, a )
R ).
(step 1. & defn of
)
a
b ( ( aRb
bRa )
a = b )
(
implication
premise is false.)
Therefore, asymmetry implies antisymmetry.Slide5
Copyright © Peter Cappello 2011
5
Exercise 20 continued
Must an antisymmetric relation be asymmetric?
(
a
b ( ( aRb
bRa )
a = b ) )
a
b ( aRb
( b, a )
R )?
Work on this question in pairs.Slide6
Copyright © Peter Cappello 2011
6
Exercise 20 continued
Must an antisymmetric relation be asymmetric ?
(
a
b ( (aRb
bRa )
a = b ) )
a
b ( aRb
(b, a)
R ) ?
Proof that the implication is false:
Let R = { (a, a) }.
R
is
antisymmetric.
R
is not
asymmetric: aRa
(a, a)
R is false.
Antisymmetry thus
does not
imply asymmetry.Slide7
Copyright © Peter Cappello 2011
7
Exercise 30
Let R = { (1, 2), (1, 3), (2, 3), (2, 4), (3, 1) }.
Let S = { (2, 1), (3, 1), (3, 2), (4, 2) }.
What is S
R?
1
2
3
4
R
S
1
2
3
4
S
RSlide8
Copyright © Peter Cappello 2011
8
Exercise 50
Let R be a relation on set A.
Show:
R is antisymmetric
R
R
-1
{ ( a, a ) | a
A }.
To prove:
R is antisymmetric
R
R
-1
{ ( a, a ) | a
A }We prove this by contradiction.R R-1 { ( a, a ) | a A }
R is antisymmetric.We prove this by contradiction.Slide9
Copyright © Peter Cappello 2011
9
Exercise 50
Prove R is
antisymmetric
R
R
-1
{ ( a, a ) | a
A }.
Proceeding by contradiction,
assume
that:
R is
antisymmetric
:
a
b ( (
aRb
bRa ) a = b ).It is not the case that
R R-1 { ( a, a ) | a A }. a b (a, b)
R R-1, where a b. (Step 1.2)aRb , where a
b. (Step 2)
aR-1
b, where a b. (Step
2)bRa
, where a b. (Step 5 &
defn of R-1)
R is not
antisymmetric, contradicting
step 1. (Steps 3 & 5)
Thus, R is
antisymmetric R
R-1 { ( a, a ) | a
A }. Slide10
Copyright © Peter Cappello 2011
10
Exercise 50 continued
Prove R
R
-1
{ ( a, a ) | a
A }
R is
antisymmetric
.
Proceeding by contradiction,
assume
that:
R
R
-1
{ ( a, a ) | a A }.R is not antisymmetric:
¬a b ( ( aRb bRa ) a = b ) a b (
aRb bRa a b ) (Step 1.2)bR-1a, where a b.
(Step 2s & defn. of R-1)( b, a ) R R-1 where a b, contradicting step 1. (Step 2 & 3)Thus, R R-1
{ ( a, a ) | a A }
R is antisymmetric.