Assignment #7
Section 3.1
11)
Drawings.
13c)
Rng(S o R) is a subset of Rng(S)
Take the domain to be the real numbers. Define S(x)=x^3 and R(x)=x^2
for x a real number. Then Rng(S) is all of the reals, but Rng(S o R)
includes only the positive real.
S o R may put a restriction on the domain of S so the Rng(S o R)
is a subset of Rng(S)
Section 3.2
1a)
{(1,2)} on A={1,2}
Reflexivity:
Take 1, (1,1) is not in the relation ==> Not reflexive
Symmetry:
Take (1,2) in the relation. (2,1) is not in the relation
==> Not Symmetric
Transitivity:
(1,2) is in the relation but there is no (2,x) in the
relation, for some x in A.
So, antecedent is false.
==> Transitive
1c)
= on N
Suppose a, b, c are natural numbers.
Reflexivity:
If a is a natural number, a = a.
==>Reflexive.
Symmetry:
If a = b, then b = a. ==> Symmetric.
Transitivity:
If a = b, and b = c, then a = c.
==> Transitive.
1d)
< on N
Suppose a, b, c are natural numbers.
Reflexivity:
If a is a natural number, a is not less than a.
==> Not Reflexive.
Symmetry:
If a < b then b > a ==> Not Symmetric.
Transitivity:
If a < b and b < c , then a < c.
==> Transitive.
1f)
<> on N
Suppose a, b, c are natural numbers.
Reflexivity:
If a is a natural number, a = a.
===> Not Reflexive.
Symmetry:
If a <> b, then b <> a. ==> Symmetric.
Transitiviy:
If a <> b, and b <> c, it does not necessarily mean that
a <> c. eg. 3 <> 4 and 4 <> 3, but 3 = 3.
==> Not Transitive.