Sample Second Midterm Key
MATH 108
Second Midterm
Sample Key
1.
a.
A relation from a set A to a set B is any subset of the Cartesian
product of A and B.
b.
Given a set A, a subset of the universe U, the characteristic function
of A is the function from U to {0,1 } given by mapping the elements of
A to 1 and the elements of UA to 0.
c.
A surjection is an onto function.
That is, it is a function whose range
is the same set as its codomain.
2.
(<=)
(a,b,c) = ((a,b), c) = ((x,y), z) = (x,y,z)
1/2 QED
(=>)
(a,b,c) = (x,y,z)
So, ((a,b), c) = ((x,y), z)
But this means that c = z.
Also, (a,b) = (x,y)
But this means that a = x and b = y.
QED
3.
a. R is not an equivalence relation, since it is not reflexive.
To see this let x elt Q.
Then x = 3^k x => k=0, but 0 not elt N.
[It is also not symmetric, as xRy => x = 3^k y => y = 3^k x, but
k not elt N.
It is transitive  see next.]
b.
Since 0 elt Z, R is reflexive by the argument above.
x = 3^k y => y = 3^k x, and hence R is symmetric.
If xRy and yRz, then x = 3^k y and y = 3^j z.
Then x = 3^k (3^j z) = 3^(k+j) z, and hence R is transitive.
Thus R is an equivalence relation.
1/2 / R = {y elt Q  1/2 R y}
= {y elt Q  1/2 = 3^ky}
= { . . . , 1/18, 1/6, 1/2, 3/2, . . . }
4.
a.
Clearly phi is a relation on phi cross phi, as the empty
set is a subset of any set, including phi cross phi.
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 Winter '10
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