CSE 260 – Homework for Lecture 8
January 29, 2003
Name ___________________________________
A. Problems from Epp Section 2.3
7. see back
8. see back
13. universal modus ponens
18. universal modus tollens
B. Translate the following from logic to English or from English to logic. Assume these definitions.
LeftOf(x, y) = x is to the left of y
Circle(x) = x is a circle
Red(x) = x is red
Behind(x, y) = x is behind y
Triangle(x) = x is a triangle
Green(x) = x is green
Large(x) = x is large
Square(x) = x is a square
Blue(x) = x is blue
Small(x) = x is small
1.
There are at least two triangles. xy(Triangle(x)  Triangle(y)  x  y)
2.
There are at most two triangles. ~xyz(Triangle(x)  Triangle(y)  Triangle(z)  x  y  x  z  z  y)
3.
There are exactly two green triangles. (answer to #1)  (answer to #2)
or xy(Triangle(x)  Triangle(y)  Green(x)  Green(y)  x  y  z((Triangle(z)Green(z))  (x=z  y=z)))
4.
There are at least two objects that are not green. xy(~Green(x)  ~Green(y)  xy)
5.
The red triangle is behind at least two circles.
xyz[Red(x)  Triangle(x)  w((Red(w)  Triangle(w)) (w=x))  Circle(y)  Circle(z)  Behind(x,y) 
Behind(x,z)  yz]
C. Write the negation of the sentence you get for #3 in Part B. Use the identities you've learned to make it so every
instance of ~ is applied to atomic sentences only. (You may use abbreviations so long as they are clear. This is a
challenging problem.)
Hint: Use DeMorgan's rules for quantifiers, DeMorgan's rules, and double negation carefully. If you have a solution you'd
like checked, email me.