document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199933>Question 1199933</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose one die is rolled. Consider the following events:\r<BR>\n" );
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document.write( "  A : the die comes up even\r<BR>\n" );
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document.write( "  B : the die comes up odd\r<BR>\n" );
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document.write( "  C : the die comes up 4 or more\r<BR>\n" );
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document.write( "  D : the die comes up at most 2\r<BR>\n" );
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document.write( "  E : the die comes up 3\r<BR>\n" );
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document.write( "a)  Are events C and E mutually exclusive? \r<BR>\n" );
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document.write( "b)  Are events B and C mutually exclusive? \r<BR>\n" );
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document.write( "c)  Are events C, D and E mutually exclusive? \r<BR>\n" );
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document.write( "d)  Are there any four mutually exclusive events among A, B, C, D and E? \r<BR>\n" );
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document.write( "e)  Are events A and B mutually exclusive?  </SPAN>\n" );
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document.write( "a) <font color=red>Yes</font>, the events C and E are mutually exclusive.\r<BR>\n" );
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document.write( "It is impossible for both events C and E to occur at the same time. In other words, set C and set E have nothing in common.<BR>\n" );
document.write( "C = {4,5,6}<BR>\n" );
document.write( "E = {3}<BR>\n" );
document.write( "This is what it means to be mutually exclusive. Another term is \"disjoint\".\r<BR>\n" );
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document.write( "If event C happens, then the die shows 4, 5, or 6. <BR>\n" );
document.write( "It's fairly clear that event E cannot happen since we don't have 3 show up. The same works in reverse as well.\r<BR>\n" );
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document.write( "b) <font color=red>No</font>, the events B and C are not mutually exclusive.\r<BR>\n" );
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document.write( "B = {1,3,5}<BR>\n" );
document.write( "C = {4,5,6}<BR>\n" );
document.write( "The overlapped or shared element is '5'\r<BR>\n" );
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document.write( "It's possible for events B and C to occur simultaneously. This is why B and C are not mutually exclusive.\r<BR>\n" );
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document.write( "c) <font color=red>Yes</font>, events C, D, and E are mutually exclusive.\r<BR>\n" );
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document.write( "C = {4,5,6}<BR>\n" );
document.write( "D = {1,2}<BR>\n" );
document.write( "E = {3}<BR>\n" );
document.write( "The three sets have nothing in common. <BR>\n" );
document.write( "They <a  rel=nofollow HREF=\"https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Applied_Discrete_Structures_(Doerr_and_Levasseur)/02%3A_Combinatorics/2.03%3A_Partitions_of_Sets_and_the_Law_of_Addition\">partition</a> the sample space {1,2,3,4,5,6} which is the set of possible outcomes of a die roll.\r<BR>\n" );
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document.write( "C union D union E = {4,5,6} union {1,2} union {3} = {1,2,3,4,5,6}\r<BR>\n" );
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document.write( "d) <font color=red>No</font>, there aren't any collection of four sets that are mutually exclusive.\r<BR>\n" );
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document.write( "There are 5 sets total to represent the 5 events.<BR>\n" );
document.write( "A = {2,4,6}<BR>\n" );
document.write( "B = {1,3,5}<BR>\n" );
document.write( "C = {4,5,6}<BR>\n" );
document.write( "D = {1,2}<BR>\n" );
document.write( "E = {3}\r<BR>\n" );
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document.write( "We're tasked to find 4 sets that are mutually exclusive. We must kick out one of those five sets.\r<BR>\n" );
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document.write( "Let's say we erased set A to be left with<BR>\n" );
document.write( "B = {1,3,5}<BR>\n" );
document.write( "C = {4,5,6}<BR>\n" );
document.write( "D = {1,2}<BR>\n" );
document.write( "E = {3}<BR>\n" );
document.write( "Earlier we found that events C,D,E were mutually exclusive, and they partition the entire sample space.<BR>\n" );
document.write( "Since these three sets union to form the entire sample space, there isn't any room for another set to be mutually exclusive to these three. <BR>\n" );
document.write( "The introduction of set B breaks the \"mutual exclusivity\".<BR>\n" );
document.write( "This means \"C,D,E\" (in any order) will not be part of a group of 4 mutually exclusive sets.\r<BR>\n" );
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document.write( "Erasing set B will lead to the same conclusion as the previous paragraph.\r<BR>\n" );
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document.write( "Let's say we erased set C<BR>\n" );
document.write( "A = {2,4,6}<BR>\n" );
document.write( "B = {1,3,5}<BR>\n" );
document.write( "D = {1,2}<BR>\n" );
document.write( "E = {3}<BR>\n" );
document.write( "But now we run into the problem of sets A and B unioning together to form the entire sample space {1,2,3,4,5,6}<BR>\n" );
document.write( "Introducing set D or E will break the \"mutual exclusivity\".\r<BR>\n" );
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document.write( "Erasing set D will lead to the same logic and conclusion as the previous paragraph.<BR>\n" );
document.write( "Erasing set E will lead to the same logic and conclusion as the previous paragraph.\r<BR>\n" );
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document.write( "This is why there aren't any collection of 4 sets that are not mutually exclusive.\r<BR>\n" );
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document.write( "e) <font color=red>Yes</font>, A and B are mutually exclusive.\r<BR>\n" );
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document.write( "A = {2,4,6}<BR>\n" );
document.write( "B = {1,3,5}<BR>\n" );
document.write( "The two sets have nothing in common.<BR>\n" );
document.write( "If a number is even, then it cannot be odd, and vice versa.<BR>\n" );
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