August 3, 1995
1. Let A be an array of integers of size 100 and let A[i] denote the i-th entry of A for a natural number i. Express the assertions given below as a proposition of a predicate logic using the predicate
. The universe is the set of natural numbers between
1 and 100. (a) Some entries of A are zero.
(b) The 6-th and 47-th entries of A are negative.
(c) If some entries of A are negative, then all the entries of A are negative.
(d) If the first 10 entries of A are nonnegative, then the last 10 entries are also nonnegative.
(e) All odd numbered entries are negative.
(f) All entries of A between the 10-th and 50-th entries are sorted in non-decreasing order.
2. Fill in the blanks with shortest strings.
(a) If P is necessary for Q, then Q is
for P. (b) The power set of {
}}} is { 
}}}}. (c) The transitive closure of the parent-child relation on a set of people is the
relation. (d) Let
be defined as 
, x R y iff 
is even. Then R is reflexive,
, and
.
Hence R is
relation. (e) If the digraph of an order relation has a cycle, it can not be a partial order because it violates the
property. (f)
. (g)
=
3. Let
, and 
.Then prove that if
, then
. 4(a) Define the set of negative odd numbers inductively.
(b) Prove by induction that
5(a) Define the following relation R on the set of natural numbers N inductively: For all natural numbers a and b,
iff 
. (b) Prove by induction that for every natural number n, if 0 < n, then
,
where the binary relation R on the set of naturals N is defined as (1)
. (2)
if 
, then 
and 
. (3) Extremal clause.