CS 281 Solution to Test I
1. For the following proposition find an equivalent proposition which uses only tex2html_wrap_inline39 and tex2html_wrap_inline41 , and simplify the resultant proposition as much as possible:
tex2html_wrap_inline43
tex2html_wrap_inline45
tex2html_wrap_inline47
tex2html_wrap_inline49

2. Fill in the blanks with the shortest string of characters so that the resultant proposition is valid.

(a) tex2html_wrap_inline51 tex2html_wrap127 ]]
(b) tex2html_wrap_inline55 [ tex2html_wrap129 tex2html_wrap_inline59
(c) tex2html_wrap_inline61 [ tex2html_wrap131 tex2html_wrap_inline65 tex2html_wrap133 ]
(Other than tex2html_wrap_inline69 ).
(d) tex2html_wrap_inline71 tex2html_wrap135 ]

3. Simplify the following proposition:
tex2html_wrap_inline75
tex2html_wrap_inline77 tex2html_wrap_inline79
tex2html_wrap_inline77 tex2html_wrap_inline83
tex2html_wrap_inline77 tex2html_wrap_inline87 tex2html_wrap_inline77 1.

4. Express the assertions given below as a proposition of a predicate logic using the following predicates. The universe is the set of integers.

A(x,y,z): x + y = z,
E(x): x is even,
G(x,y): x > y,

(a) Every integer is greater than 1, if it is even and positive.
tex2html_wrap_inline107
(b) There is no odd integer which is greater than any integer.
tex2html_wrap_inline109
OR tex2html_wrap_inline111
(c) There is an even integer which is greater than any nonnegative odd integer.
tex2html_wrap_inline113
(d) If an even integer is added to an odd integer, then the result is odd.
tex2html_wrap_inline115

(e) For an integer to be odd, it is necessary that adding an odd integer to it produces an even integer.
tex2html_wrap_inline117
(f) An integer is greater than 1 only if it is positive.
tex2html_wrap_inline121

5. Find the converse and the contrapositive of question 4(f) above. State them in English.
Converse: If an integer is positive, then it is greater than 1.
Contrapositive: If an integer is not positive, then it is not greater than 1.




S. Toida
Thu May 29 11:03:49 EDT 1997