Question 1208744: It is estimated that 30 percent of all adults in the United States are obese and that 3 percent suffer
from diabetes. If 2 percent of the population both is obese and suffers from diabetes, what
percentage of the population either is obese or suffers from diabetes?
Found 3 solutions by ikleyn, math_tutor2020, greenestamps:
Answer by ikleyn(52803)   (Show Source):
You can put this solution on YOUR website! .
It is estimated that 30 percent of all adults in the United States are obese
and that 3 percent suffer from diabetes. If 2 percent of the population
both is obese and suffers from diabetes, what percentage of the population
either is obese or suffers from diabetes?
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We are given three subsets in the set of adults in the US:
- subset of those suffering obese (30%) (the subset O);
- subset of those suffering diabetes (3%) (the subset D).
- subset of those suffering both obese and diabetes (2%),
which is the intersection of the first two subsets (O ∩ D).
(1) From the problem
- the percent of all adult in the US that are obese but do not suffer diabetes
is 30% - 2% = 28% ( subset (O \ (O ∩ D)) );
- the percent of all adults in the US that suffer of diabetes but are not obese
is 3% - 2% = 1% ( subset (D \ (O ∩ D)) );
(2) These subsets (O \ (O ∩ D)) and (D \ (O ∩ D)) are disjoint.
(3) The problem asks about the percent of the union of these two subsets.
Since the subsets are disjoint, the desire percent is the sum 28% + 1% = 29%. ANSWER
Solved.
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Tutor @math_tutor2020 came with his own interpretation of the term "either-or",
treating it as "OR inclusive". This interpretation IS NOT CORRECT.
The correct and commonly / (uniquely) accepted interpretation of "either-or"
in problems like this one in the post is "OR exclusive". And no any other.
See this link (AI overview)
https://www.google.com/search?q=either+or+in+Math+neaning&rlz=1C1CHBF_enUS1071US1071&oq=either+or+in+Math+neaning&gs_lcrp=EgZjaHJvbWUyBggAEEUYOTIJCAEQIRgKGKABMgkIAhAhGAoYoAEyCQgDECEYChigATIJCAQQIRgKGKAB0gEKMTMzNjRqMGoxNagCCLACAQ&sourceid=chrome&ie=UTF-8
In math, " either or " signifies a choice between two options, meaning that only one
of the two possibilities can be true at a time, but not both.
Therefore, his solution to the given problem in the post IS INCORRECT / IRRELEVANT.
//////////////////////////////
Regarding the comment by @greenestamps, my impression is that he
did not read my solution to the end, at all.
His counter-argument is that
Being obese and suffering from diabetes are NOT two disjoint things; a person CAN be both.
My solution and my arguments literally follow to this idea.
In this problem, the obese and the diabetes are the essences of the same row and of the same level,
so "either-or" in this case can be and should be interpreted as "OR-exclusive".
Would the problem want to ask "OR-inclusive", it should use simple "OR", without any "either", in this case.
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I must admit that the usage of "either-or" in English is a total mess.
While it is normal in everyday life (nobody will die, or sick, or injured,
or judged and convicted, I hope), it is unacceptable in mathematical texts.
Therefore, it is the responsibility of problem composers to write in a clear way
and do not create ambiguity.
But, although the rules of usage "either-or" in Math do exist and can be found (after long searches),
nevertheless, the contemporary Internet problem composers, in mass, do not know these rules and do not follow them.
The same is true for thousands of Math tutors that think that they are experts in English,
without having read professional Math books and Math articles, written and edited by professional Math writers.
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
OK, I continued my searches in the Internet and communications with the Google-AI.
I used keywords "using either-or in Mathematics".
Below is another explanation given by this AI.
https://www.google.com/search?q=using+either-or+in+Math&rlz=1C1CHBF_enUS1071US1071&oq=using+either-or+in+Math&gs_lcrp=EgZjaHJvbWUyBggAEEUYOTIHCAEQIRigATIHCAIQIRigATIHCAMQIRigATIHCAQQIRigAdIBCjExOTAzajBqMTWoAgiwAgE&sourceid=chrome&ie=UTF-8
- - START of the AI message - -
In mathematics, "either-or" signifies a situation where only one of two possible options
can be true at a time, typically represented by the logical "OR" symbol (∨),
meaning that a statement is true if either one of the conditions is met,
but not necessarily both; essentially, it represents the idea of choosing one option
from a set of two, excluding the possibility of both being true simultaneously.
Key points about "either-or" in math:
Exclusive nature:
Unlike in everyday language where "either-or" might sometimes imply the possibility of both being true,
in mathematics, it strictly means only one option can be true at a time.
- - END of the AI message - -
Comparing it with the points by @math_tutor2020 and @greenestamps, I conclude from this that
*****************************************************************
NEITHER @math_tutor2020 NOR @greenestamps know
how to use "either-or" correctly in Mathematics.
*****************************************************************
I do not speak about using either-or in everyday life - it is total mess.
I speak here specifically about using "either-or" in Math, which is a very special case,
different from its using in everyday life.
*********************************************************************
For anybody, who wants to know TRUTH, my detailed description
of using either-or in Math is below.
*********************************************************************
Everybody knows that "either-or" can work as "OR-exclusive" or "OR-inclusive",
but not all know in which situation to apply and to use them correctly.
(A) If "either A or B" is used for disjoint sets/events A and B (when the intersection
is empty), then it means "take any element from A or from B", i.e. selection goes
from the union of these sets/events.
It is the case of "OR-inclusive".
The meaning of "either-or" in this case is the same as single "OR".
(B) If "either A or B" is used for sets/events A and B (when the intersection is not empty),
then it means "for any element from A or from B, but not from their intersection",
i.e. selection goes from the union of these sets/events MINUS the intersection.
It is the case of "OR-exclusive".
The meaning of "either-or" in this case is "take a single element from A or from B,
but not from the intersection".
In this case, the meaning of "either-or" is different from traditional single "or".
Again,
(C) If "either-or" is used for sets A and B having different descriptions, which people
consider as different kind of essences, then "either-or" works as "OR-inclusive" -
the selection of single item from A or B goes from the union.
It is the case of "OR-inclusive".
The meaning of "either-or" in this case is the same as single "OR".
(D) If "either-or" is used for sets A and B having similar descriptions, which people
consider as essences of the same row or of the same level, then "either-or" works
as "OR-exclusive" - the selection of a single item from A or B goes from (A U B) \ (A ∩ B).
In this case, the meaning of "either-or" is different from traditional single "or".
In cases (A) and (C), "either-or" can be used, but its using is excessive,
since these cases (A) and (C) are fully covered by single "OR",
which never creates misunderstanding.
The real cases, when "either-or" is really applicable according to its meaning/essence,
are cases (B) and (D).
And, probably, the last my notice.
These rules of using "either-or" are not a subject to discuss or to argue.
They are the rules to follow. It is the Math English language with its strict rules.
Answer by math_tutor2020(3817)  (Show Source):
You can put this solution on YOUR website!
Let's say hypothetically there are 1,000,000 adults in the US.
30% of which are obese, so 0.30*(1,000,000) = 300,000 adults are obese.
3% have diabetes, so 0.03*(1,000,000) = 30,000 adults have diabetes.
2% have both conditions, 0.02*(1,000,000) = 20,000 adults are obese and diabetic.
A = set of people who are obese
B = set of people who are diabetic
n(A or B) = n(A) + n(B) - n(A and B)
n(A or B) = 300,000 + 30,000 - 20,000
n(A or B) = 310,000 people are obese, diabetic, or both.
(310,000)/(1,000,000) = 0.31 = 31% of the population is obese, suffers from diabetes, or both.
Answer: 31%
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
Ignore the pompous (and incorrect) comment from tutor @ikleyn saying that the response from the other tutor is "INCORRECT/IRRELEVANT" (my quotation marks).
If a car is either red or black, then that "either - or" obviously means one or the other and not both. But if a car is either red or a Ford, then it can be both.
Being obese and suffering from diabetes are NOT two disjoint things; a person CAN be both.
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