a Can you find a counterexample to show that it is not an onto map? Try some pairings.
B
b Can you find a counterexample to show that it is not an onto map? Try some pairings.
C
c Try some pairings. What can you conclude?
D
d Can you find a counterexample to show that it is not an onto map? Try some pairings.
A
aInto
B
bInto
C
cOnto
D
dInto
Practice makes perfect
a Let's start by reviewing what the set of whole numbers is. Recall that whole numbers are those we use to count, including 0.
Whole Numbers [0.8em]
{ 0, 1, 2, 3, ... } 0.1cm In this case, the function is pairing each number with its double. We can show that this mapping does not map the whole numbers onto the whole numbers by counterexample. Think of the first pairings. 0 will be paired to 0, 1 with 2, and so on. However, there is no number that will be paired with 1, 3,etc, in the outputs.
Therefore, this function maps the set of the whole numbers into the set of whole numbers, but notonto the set of whole numbers.
b In this case, the function is pairing each number to one more than it. We can show that this mapping does not map the whole numbers onto the whole numbers by counterexample. Think of the first pairings. 0 will be paired to 1, 1 with 2, and so on. However, there is no number that will be paired with 0 in the outputs.
Therefore, this function maps the set of whole numbers into the set of whole numbers, but notonto the set of whole numbers.
c In this case, the function is pairing each number with itself. We can do the first pairings to get an idea on how this mapping behaves.
Notice that as every number maps to itself, all of the output whole numbers will be used. Therefore, this is function maps the whole numbers onto the whole numbers.
d In this case, the function is pairing each number to its square. We can show that this mapping does not map the whole numbers onto the whole numbers by counterexample. Think of the first pairings. 0 will be paired to 0, 1 with 1, 2 with 4 and so on. However, there is no number that will be paired with 2, 3,etc, in the outputs.
Therefore, this function maps the set of whole numbers into the set of whole numbers, but notonto the set of whole numbers.
