CS 390 Test I
February 17, 1994
1. Let A and B be a set and f: 
be a function.
Also let S and T be subsets of A.
Then is the set 
a subset of 
?
Justify your answer, that is either prove or give a counterexample. [20]
2. A relation 
is defined as
for any sets A and B, A B if and only if 
.
Which of the properties, reflexive, symmetric and transitive hold for ?
Briefly give your reasons. [20]
3. Translate the following English sentences into a logical statement
using the parameters given inside ( )and only those.
(a) For 
, x < -2 is sufficient. (universe: all integers,
: an integer x is greater than or equal to an integer y.) [7]
(b) For 
, 
is necessary. (Same as (a)) [7]
(c) Not all integers are even. (universe:all reals i.e. fractions and
integers, E(x): x is an even number, I(x): x is an integer.) [6]
4. Describe in English the language L defined below.
Briefly justify your answer. [20]
(1) 
(2) If 
, then 
, 
, and 
.
(3) Nothing is in L unless it is obtained from (1) and (2) above.
5. Prove the following by induction. [20]
The language L defined below is the set of strings which have one or more 'a'
followed by any number of 'b'.
Definition of L:
(1)
(2) If , then 
and .
(3) Nothing is in L unless it is obtained from (1) and (2) above.
S. Toida
Tue Jan 14 10:45:47 EST 1997