1. Find the powerset of each of the following sets: [9]
(a) {1}
{
, {1} }
(b) {
, {
}}
{ ,
{}, { {
} }, {
, {
}} }
(c)
{
}
2. Find the following Cartesian products: [9]
(a) {
{
}
(b) {
{
}
(c)
3. Indicate which of the following are true and which are false. [15]
(a)
False
(b)
False
(c)
True
(d) {
False
(e)
True
4. Find the smallest four elements of the set S defined recursively as follows: [8]
Basis Clause:
Inductive Clause: If ,
then
.
Extremal clause: Nothing is in S unless it is obtained from the above two clauses.
1, 4, 13, 40
5. Recursively define each of the following sets:
(a) The set of non-negative integers that produce remainder 1 when divided by 5. [10]
Let F be the set to be defined.
Basis Clause:
Inductive Clause: If ,
then
.
Extremal Clause: Nothing is in F unless it is obtained by the Basis and Inductive Clauses.
(b) {
is a natural number.} [12]
Let T be the set to be defined.
Basis Clause:
Inductive Clause: If ,
then
.
Extremal Clause: Nothing is in T unless it is obtained by the Basis and Inductive Clauses.
6. Prove the following statements on sets A and B.
(a)
[10]
Since
and
,
by applying formula # 7 on properties of set operations
( If
and
,
then
),
.
But
.
Hence
.
(b) If
,
then
.
[13]
Since
,
for an arbitrary x,
.
Hence
or .
From this, if ,
then by disjunctive syllogism .
Hence for an arbitrary x if ,
then .
Hence
.
7. Which rules of inference are used to establish the conclusion of the following argument ? [14]
Premises:
"All lions are fierce."
"Some lions do climb trees."
Conclusion:
"There are fierce creatures that climb trees."
Give your answer using L(x), C(x) and F(x) to denote "x is a lion",
"x climbs trees"
and "x is fierce", respectively, and assuming the universe is the set
of all creatures. Using these symbols, the statements given above
can be expressed as follows:
---------------------
Answer:
(1)
: premise
(2)
: premise
(3)
for some creature c : Existential Instantiation on (2)
(4) L(c) : Simplification on (3)
(5)
: Simplification on (3)
(6)
: Universal Instantiation on (1)
(7) F(c) : Modus Ponens on ((4) and (6)
(8)
: Conjunction on (5) and (7)
(9)
: Existential Generalization on (8)
------------------------------------------------
You may use the following table.
Logical Equivalences and Implications -- Omitted.
Additional Inference Rule: Conjunction
P
Q
------------------
P
Q