Find the union and the intersection of the two sets and where
(a)
(b)
(c)
Solution (c):
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikibooks.org/v1/":): {\displaystyle A_1\cap A_2=\{(x,y):1<x<2, 0<y<2\}}
List all the possible arrangements of the four letters m,a,r and y. Let be the collection of the arrangements in which y is in the last position. Let be the collection of the arrangements in which m is in the first position. Find the union and intersection of and .
To join a certain club, a person must be either a statistician or a mathematician or both. Of the 25 members in this club, 19 are statisticians and 16 are mathematicians. How many persons in the club are both a statistician and a mathematician?
Soluion: The number of persons which are statisticians or mathematicians = number of statisticians + number of mathematicians - number of persons which are both statisticians and mathematicians. This can be proved directly using properties of the counting measure. The general proof is as follows: Let A and B be two finite sets. Endow the natural numbers with the counting measure which we shall denote by n. We shall denote union by + and intersection by . and the context will make it clear whether we mean actual addition, multiplication or union, intersection. Now A-B (Here - represents the relative complement, AB and B-A are disjoint and as n is sigma additive so
n(A+B)=n(A-B)+n(B-A)+n(AB)
n(A)=n(A-B)+n(AB) which implies n(A-B)=n(A)-n(AB)
n(B)=n(B-A)+n(AB) which implies n(B-A)=n(B)-n(AB)
Substituting n(A-B) and n(B-A) from (2) and (3) into (1) gives us n(A+B)=n(A)+n(B)-n(AB).
So the answer is 10.
After a hard fought football game, it was reported that of the 11 starting players, 8 hurt a hip, 6 hurt an arm, 5 hurt a knee, 3 hurt both a hip and an arm, 2 hurt both a hip and a knee, 1 hurt both an arm and a knee, and no one hurt all three. Comment on the accuracy of the report.
Solution: We shall first establish that in the language of the previous problem:
.
Now,
Now if we let A, B and C to be the set of players with injured hips, knees and arms respectively, we would find the contradiction on substituting the appropriate terms in the above formula. Hence the report is not accurate.