HW

1
HW6

• Ex 3.2
• 2, 3, 7, 8
• Ex 3.3
• 1, 6

2
Ex3.2-2

• Whats wrong with the following proof.
• Assume relation R is symmetric and transitive.
  Then xRy ? yRx.
• Since xRy and yRx, then by transitivity, we have
  xRx. Therefore R is reflexsive.
• xRy?yRx? xRy and yRx? ??.

3
Ex3.2-3

• A) is brother of
• Not reflexive, symmetric, transitive,
  irreflexive, not antisymmetric
• B) has the same birthday
• Reflexive, symmetric, transitive, not
  irreflexive, not antisymmetric
• C)
• Reflexive, not symmetric, not transitive, not
  irreflexive, not antisymmetric
• D) is a multiple of
• Reflexive, not symmetric, transitive, not
  irreflexive, antisymmetric

4
Ex3.2 -7

• Determine which of the relations in Ex 3 are
  partial orderings and which are linear ordering
• Partial ordering reflexive, symmetric,
  transitive
• ? D)
• Linear ordering antisymmetric, transitive,
  trichotomy
• ? no

5
Ex3.2-8

• Find the transitive closure of the relation shown
  in Fig.
• R(a,b), (a,c), (a,d), (b,b), (b,d), (c,c),
  (c,b), (d,b)
• R1R
• RR È (a,d), (c,d), (d,d)

a
b
c
d
6
Ex3.3-1

• A)
• B)

RS
1 2 3 4
1 2 3 4
SR
1 2 3 4
1 2 3 4
7
Ex3.3 - 1

• c)
• d)

RR
1 2 3 4
1 2 3 4
SS
1 2 3 4
1 2 3 4
8
Ex3.3-6

• Show that a relation R on a set A is symmetric
  iff R R-1
• If R on A is symmetric, R R-1
• if xRy, then xR-1y. (If xRy, then yRx.
  (symmetric))
• If xR-1y, then xRy. (if xR-1y, then yRx)
• If RR-1, R is symmetric.
• If xRy, then yRx. (if xRy, then xR-1y. )