Question 1097303
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U = universal set


Let's say that the universal set U has p items inside it where p is some positive integer


*[Tex \Large U = \{a_{1},a_{2},\ldots, a_{p}\}]


Define positive integers m and n such that {{{m <= n <= p}}}


Set A has n items inside it


*[Tex \Large A = \{a_{1},a_{2},\ldots, a_{n}\}]


while set B has m items inside it


*[Tex \Large B = \{a_{1},a_{2},\ldots, a_{m}\}]


Every item inside set B is also found in set A. This makes set B a subset of set A.


The probability of selecting an item from set A is {{{n/p}}} since we have n items from set A and p items from set U.


The probability of selecting an item from set B is {{{m/p}}} since we have m items from set B and p items from set U.


Because {{{m <= n}}}, this means that {{{m/p <= n/p}}}. 
We can divide both sides of the first inequality ({{{m <= n}}}) by p to get the second inquality ({{{m/p <= n/p}}}). 
The inequality sign will not flip because p is a positive number.


So this proves that P(B) <= P(A) is true, since,
P(A) = n/p
P(B) = m/p


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