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Sets: Mutually exclusive and collectively exhaustive
1. Sets: Mutually exclusive and collectively exhaustive
Items in a set can be partitioned into categories in many ways.
Given a set:
There is a question or relation.
There are answers or equivalence sets (without an ordering)
Mutually exclusive means that "yes" or a "no" are distinct and cannot be both at the same time.
Collectively exhaustive means that everything must be either a "yes" or a "no". There are no other possibilities.
| Question |
Possible |
Answers |
| Relation |
Equivalence |
Sets |
| Is it true? |
yes |
no |
| What sex are you? |
male |
female |
| Who do you serve? |
God |
mammon |
2. Mutually exclusive
Mutually exclusive means that no person can be on both sides. There is no non-empty intersection of the two groups.
3. Collectively exhaustive
Collectively exhaustive means that each person is included in the division. There is nothing outside of these two groups.
4. Mutual exclusion
A partition of a set of items into categories is
mutually exclusive if each item in the set can belong to no more than one category.
One meaning of the word "mutually" is "shared by two more".
One meaning of the word "exclusive" is "restricted in use".
If you cannot walk or chew gum at the same time, then "
walking" and "
chewing" gum "
are mutually exclusive categories for you". That is, you cannot be doing both at the same time.
Are "winning" and "losing" mutually exhaustive categories for basketball games at your school?
Are "heads" and "tails" mutually exclusive categories for the result of flipping a coin?
Are "playing" and "winning" mutually exclusive categories at your school for basketball games?
5. Collective exhaustion
A partition of a set of items into categories is
collectively exhaustive if each item in the set must belong to at least one category.
One meaning of the word "collectively" is "collected together".
One meaning of the word "exhaustive" is "complete"
Are "winning" and "losing" collectively exhaustive categories for basketball games at your school?
Are "heads" and "tails" collectively exhaustive categories for the result of flipping a coin?
6. Both restrictions
A partition of a set of items into categories is both mutually exclusive and collectively exhaustive if each item in the set belongs to one and only one category.
7. All possibilities
neither mutually exclusive nor collectively exhaustive
both mutually exclusive and collectively exhaustive
mutually exclusive but not collectively exhaustive
collectively exhaustive but not mutually exclusive
8. Diagram
In text diagram terms:
. | Collectively
. | Exhaustive
. |-------------
. | No | Yes
-----------------------------
Mutually | No |
Exclusive |Yes |
9. End of page
