A group of college students were asked
about their TV watching habits.
Out of these surveyed, 39 watch "The Walking Dead,"
32 watch "The Blacklist,"
and 29 watch "Game of Thrones."
Additionally, eight watch "The Walking Dead"
and "The Blacklist,
12 watch "The Walking Dead" and "Game of Thrones,"
and 11 watch "The Blacklist" and "Game of Thrones."
There are eight students who watch all three shows.
How many watch at least one of the shows?
If we let set W contain the students
that watch "The Walking Dead,"
set B contain the students that watch "The Blacklist,"
and set G contain the students that watch "Game of Thrones,"
we can determine how many watch at least one of the shows
by determining the cardinality of W union B union G.
So we could answer this question using
the formula shown below,
or we can use the Venn diagram.
In this lesson, we will cover both methods.
Let's first write out all the given information
in relation to sets W, B, and G.
Because 39 watch "The Walking Dead,"
the cardinality of W is equal to 39.
Since 32 watch "The Blacklist,"
the cardinality of B is equal to 32.
And since 29 watch "Game of Thrones,"
the cardinality of G is equal to 29.
Next, eight watch "The Walking Dead" and "The Blacklist,"
which means the cardinality
of W intersect B is equal to eight.
And since 12 watch "The Walking Dead" and "Game of Thrones,"
the cardinality of W intersect G
is equal to 12.
And since 11 watched "The Blacklist" and "Game of Thrones,"
the cardinality of B intersect G is equal to 11.
We also know eight watch all three,
which indicates a cardinality
of W intersect B intersect G
is equal to eight.
So again, because we're looking
for how many watch at least one of the shows,
meaning how many are in set W,
or set B, or set G,
we're trying to determine the cardinality
of W union B union G,
which we know from the formula below
which deals with the cardinality of the union of three sets.
The cardinality of this union is equal
to the cardinality of W
plus the cardinality of B
plus the cardinality of G
minus the cardinality of W intersect B
minus the cardinality of W intersect G
minus the cardinality of B intersect G,
and then finally,
plus the cardinality of W intersect B intersect G.
And we have all of these values above.
The cardinality of W union B union G
is equal to the cardinality of W, which is 39,
plus the cardinality of B, which is 32,
plus the cardinality of G, which is 29,
and then minus the cardinality
of W intersect B, which is eight,
minus the cardinality of W intersect G, which is 12,
minus the cardinality of B intersect G, which is 11,
and finally, plus the cardinality
of W intersect B intersect G, which is eight.
Simplifying, we get the cardinality
of W union B union G
is equal to 77.
This indicates 77 students watch at least one of the shows.
But now, let's go ahead and verify this
by, again, putting the Venn diagram.
To complete the Venn diagram,
we'll actually work our way backwards starting with,
we know that eight students watch all three shows,
and therefore there's an eight in this region here,
the intersection of W, B, and G.
Next, we know 11 students watch "The Blacklist"
and the "Game of Thrones,"
and therefore there must be 11 students
in the intersection of B and G, this region here,
but these eight students are part of the 11,
and therefore there's three students in this region here.
Next, 12 students watch "The Walking Dead"
and the "Game of Thrones,"
which means there are 12 students
in the intersection of W and G, this region here,
but these eight students are part of the 12,
and therefore there are four students here
that only watch "The Walking Dead" and "Game of Thrones."
And we also know eight students watch "The Walking Dead"
and "The Blacklist," which means there are eight students
in the intersection of W and B, this set here,
but these eight students are the eight students given
that watch "The Walking Dead" and "The Blacklist,"
and therefore, there are zero students
that only watch "The Walking Dead" and "The Blacklist."
Continuing backwards, we now know
29 watch "Game of Thrones."
There must be 29 students in set G,
but there's already 12 plus three or 15 students in set G.
29 minus 15 is 14.
14 students only watch "Game of Thrones."
32 watch "The Blacklist,"
which indicates there must be 32 students in set B.
There's already 11 students in set B.
32 minus 11 is 21.
21 students only watch "The Blacklist."
And finally, 39 students watch "The Walking Dead,"
but there's already 12 students in set W.
39 minus 12 is equal to 27.
So let's verify we get the same result
for the cardinality of W union B union G,
which is equal to,
again, the number of elements
in the union of the three sets,
which is this set here.
Let's be careful adding these up.
We have 27 plus 21
plus four plus eight plus three
and then plus 14.
And we also get 77.
So again, this verifies 77 students watch
at least one of the three shows.
I hope you found this helpful.