COT 3100 Homework #4 Fall ’00
Lecturer: Arup Guha
Assigned: 10/17/00
Due: 10/24/00 (No late assignments)
1)
Prove or disprove: If a relation R, defined as a subset of A x A, is transitive, then R
°
R
is transitive as well.
We must show that if (a,b)
∈
R
°
R
∧
(b,c)
∈
R
°
R, that (a,c)
∈
R
°
R.
If we have (a,b)
∈
R
°
R, by the defn. of relation composition, there exists a d such that
(a,d)
∈
R
∧
(d,b)
∈
R. We also know that R is transitive. Hence we can deduce that
(a,b)
∈
R.
If we have (b,c)
∈
R
°
R, by the defn. of relation composition, there exists a e such that
(b,e)
∈
R
∧
(e,c)
∈
R. We also know that R is transitive. Hence we can deduce that
(b,c)
∈
R.
But, if we have that (a,b)
∈
R
∧
(b,c)
∈
R, by defn. of function composition, we know
that (a,c)
∈
R, as desired. Thus, R
°
R is transitive.
2)
Define sumdigits(a), where a is a positive integer, to be the sum of the digits in the
decimal representation of the positive integer a. Using this definition of the function
sumdigits, consider the following relation:
R = {(a,b) | a
∈
Z
+
∧
b
∈
Z
+
∧
(a
≥
b
∨
sumdigits(a)
≥
sumdigits(b)}
Determine if this relation is (i)reflexive, (ii)irreflexive, (iii)symmetric, (iv)antisymmetric,