Answer:
when
.
Step-by-step explanation:
The question states that is proportional to
. In other words, there is a constant
(
) such that
for all
.
The question also states that when
. Make use of this equality to find the value of
.
Since and
, it must be true that
. Substitute
and
into the equation
and solve for
:
.
.
Thus, for all
.
Substitute in the value to find the corresponding value of
:
.
Part (a)
Answer: See the attached image below to see the filled out chart.
Note how rational and irrational numbers have nothing in common. This means there is no overlap. So they go in the rectangles. The two sets of numbers join up to form the entire set of real numbers.
Integers are in the set of rational numbers. This is because something like 7 is also 7/1; however 1/7 is not an integer. So not all rational numbers are integers. The larger purple circle is the set of integers.
The smaller blue circle is the set of whole numbers. The set of whole numbers is a subset of integers. Recall the set of whole numbers is {0,1,2,3,...} so we ignore the negative values only focusing on 0 and positive numbers that don't have any fractional values. In contrast, the set of integers is {..., -3, -2, -1, 0, 1, 2, 3, ...} here we do include the negatives.
============================================================
Part (b)
1) Some irrational numbers are integers.
This is false. An irrational number is not rational. The set of integers is contained entirely in the set of rational numbers.
---------------------------------
2) Some whole numbers are not irrational numbers.
This is false. The statement implies that some whole numbers are irrational, but the set of whole numbers is inside the set of rational numbers, which has no overlap with the irrationals. The statement should be "All whole numbers are not irrational numbers".
---------------------------------
3) All rational numbers are whole numbers.
This is false. A rational number like 1/3 is not a whole number.
---------------------------------
4) All integers are whole numbers
This is false. An integer like -44 is not a whole number because the set of whole numbers is {0,1,2,3,...} and we're not including negative values.
---------------------------------
In summary, all four statements for part (b) are false.
