Remember that a function from to has the property that each element of has been assigned to exactly one element of .
Why is not a function from to if
a)
b)
c)
a) is not defined for any value.
b) Negative square roots are undefined, or complex numbers (not real numbers).
c) There are multiple values in the domain that map to single values of the codomain.
the domain is the set of possible inputs for which the function is defined, and the range is the
set of all possible outputs on these inputs.
Find the domain and range of these functions.
a) the function that assigns to each pair of positive integers
the maximum of these two integers
b) the function that assigns to each positive integer the
number of the digits that do
not appear as decimal digits of the integer
c) the function that assigns to a bit string the number of
times the block appears
d) the function that assigns to a bit string the numerical
position of the first in the string and that assigns the
value to a bit string consisting of all s
a) The domain is the set of ordered pairs (, ) and the range is .
b) The domain is . The base 10 representation of an integer has to have at least one digit, and at most nine digits do not appear, so the number of missing digits could be any number less than 9. Thus the range is .
c) The domain is the set of bit strings and the range is (set of natural numbers)
d) The domain is the set of bit strings and the range is (if the string contains no s then the value is by definition, or its position could be .
A function is onto when its range is the entire codomain.
Determine whether the function is onto if
a)
b)
c)
d)
e)
a) Given any integer , so this is onto.
b) No, the range contains no negative integers.
c) Yes this is onto (similar to reasoning for a)
d) No, similar to reasoning for b(no neg ints)
e) Yes, maps all integers
(remember a function is bijective when each value of the domain is mapped to the codomain exactly once)
One way to determine whether a function is a bijection is to try to construct its inverse.
Determine whether each of these functions ia bijection from to
a)
b)
c)
d)
a) yes
b) no ( not injective, the codomain has no negative
c) this is a bijection since it has an inverse function.
d) No, and have the same image, for all .
Let and let for all . Show that is strictly decreasing if and only if the function is strictly increasing.
This proposition is biconditional so we have to prove the proposition both ways. Suppose is strictly decreasing. This means that whenever . To show that is strictly increasing, suppose that . Then . Conversely, suppose that is strictly increasing. This means that whenever . To show that is strictly decreasing, suppose that . Then .