document.write( "Question 1203553: Theorem 1.4.3 is
\n" ); document.write( "P(AuB)=P(AnB')+P(B)
\n" ); document.write( "= [P(A) - P(A n B)] + P(B)
\n" ); document.write( "= P(A) + P(B) - P(A n B)\r
\n" ); document.write( "\n" ); document.write( "Theorem 1.4.4 is
\n" ); document.write( "P(A u B u C) = P(A) + P(B) + P(C)
\n" ); document.write( "- P(A n B) - P(A n C) - P(B n C)
\n" ); document.write( "+ P(A n B n C)\r
\n" ); document.write( "\n" ); document.write( "Prove Theorem 1.4.4. Hint: Write A u B u C = (A u B) u Cand apply Theorem 1.4.3. \n" ); document.write( "

Algebra.Com's Answer #839219 by ikleyn(52800) $\"\"$ $\"About$

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document.write( "In order to understand the formula of inclusion-exclusion principle,\r\n" );
document.write( "it is very useful to keep in mind this simple reasoning.\r\n" );
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document.write( "When we calculate the sum \r\n" );
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document.write( "    P(A) + P(B) + P(C) for P(A U B U C),     (1)\r\n" );
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document.write( "it seems very natural and does not arouse suspicion - so, it looks as a good first approximation.\r\n" );
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document.write( "But thinking longer, you understand that every part  P(A n B),  P(A n C)  and  P(B n C)\r\n" );
document.write( "you count twice in this sum  P(A) + P(B) + P(C).\r\n" );
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document.write( "Therefore, next step is to subract  P(A n B) + P(A n C) + P(B n C)  from the sum   P(A) + P(B) + P(C).\r\n" );
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document.write( "So, you get then  P(A) + P(B) + P(C) - P(A n B) - P(A n C) - P(B n C).     (2)\r\n" );
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document.write( "It is good as the next, second approximation.\r\n" );
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document.write( "But thinking further, you understand that in expression  P(A) + P(B) + P(C) - P(A n B) - P(A n C) - P(B n C)\r\n" );
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document.write( "the part P(A n B n C) is added three times in the first three addends and subtracted three times\r\n" );
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document.write( "in the next three terms. So, now this part  P(A n B n C)  simply ABSENTS in the second approximation (2).\r\n" );
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document.write( "THEREFORE, you MUST add  P(A n B n C)  to (2),  and after doing it, you get  \r\n" );
document.write( "final formula of the Inclusion-Exclusion principle\r\n" );
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document.write( "    +-------------------------------------------------------------------------+\r\n" );
document.write( "    |   P(A U B U C) = P(A) + P(B) + P(C) - P(AB) - P(AC) - P(BC) + P(ABC).   |\r\n" );
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document.write( "You may consider it as a formal or informal proof of the formula.\r\n" );
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\r
\n" ); document.write( "\n" ); document.write( "As soon as you got this reasoning and placed it in your mind,
\n" ); document.write( "you do understand the Inclusion-Exclusion principle in whole.\r
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