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P(A|B) may or may not be equal to P(A), i.e., the unconditional probability or absolute probability of A. If P(A|B) = P(A), then events A and B are said to be independent: in such a case, knowledge about either event does not alter the likelihood of each other. P(A|B) (the conditional probability of A given B) typically differs from P(B|A).
A p-complement is a complement to a Sylow p-subgroup. Theorems of Frobenius and Thompson describe when a group has a normal p-complement. Philip Hall characterized finite soluble groups amongst finite groups as those with p-complements for every prime p; these p-complements are used to form what is called a Sylow system.
In set theory, the complement of a set A, often denoted by (or A′), [1] is the set of elements not in A. [ 2 ] When all elements in the universe , i.e. all elements under consideration, are considered to be members of a given set U , the absolute complement of A is the set of elements in U that are not in A .
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We can use Boole's Inequality to solve this problem. By finding the complement of event "all five are good", we can change this question into another condition: P( at least one estimation is bad) = 0.05 ≤ P( A 1 is bad) + P( A 2 is bad) + P( A 3 is bad) + P( A 4 is bad) + P( A 5 is bad) One way is to make each of them equal to 0.05/5 = 0.01 ...
In this Boolean algebra, union can be expressed in terms of intersection and complementation by the formula = (), where the superscript denotes the complement in the universal set . Alternatively, intersection can be expressed in terms of union and complementation in a similar way: A ∩ B = ( A ∁ ∪ B ∁ ) ∁ {\displaystyle A\cap B ...
c is a strong negator (aka fuzzy complement). A function c satisfying axioms c1 and c3 has at least one fixpoint a * with c(a *) = a *, and if axiom c2 is fulfilled as well there is exactly one such fixpoint. For the standard negator c(x) = 1-x the unique fixpoint is a * = 0.5 . [2]
The Frobenius normal p-complement theorem shows that if every normalizer of a non-trivial subgroup of a Sylow p-subgroup has a normal p-complement then so does G.For applications it is often useful to have a stronger version where instead of using all non-trivial subgroups of a Sylow p-subgroup, one uses only the non-trivial characteristic subgroups.