7.01b Set notation: basic language and notation of sets, partitions

3 The diagram shows three sets, A, B and C. Each region of the diagram contains at least one element. The diagram shows that B is a subset of \(\mathrm { A } , \mathrm { C }\) is a subset of A , and that B shares at least one element with C . \includegraphics[max width=\textwidth, alt={}, center]{7330fba5-720f-47d5-a2ac-ad24e0cf097b-3_410_615_342_726} The two graphs below give information about the three sets \(\mathrm { A } , \mathrm { B }\) and C . The first graph shows the relation 'is a subset of' and the second graph shows the relation 'shares at least one element with'. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Draw two graphs to represent the sets \(\mathrm { X } , \mathrm { Y }\) and Z shown in the following diagram. \includegraphics[max width=\textwidth, alt={}, center]{7330fba5-720f-47d5-a2ac-ad24e0cf097b-3_415_613_1388_731}
2. Draw a diagram to represent the sets \(\mathrm { P } , \mathrm { Q }\) and R for which both of the following graphs apply. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{7330fba5-720f-47d5-a2ac-ad24e0cf097b-3_202_264_1980_621} \captionsetup{labelformat=empty} \caption{'is a subset of'}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{7330fba5-720f-47d5-a2ac-ad24e0cf097b-3_200_260_1982_1155} \captionsetup{labelformat=empty} \caption{'shares at least one element with'}
   \end{figure}

7

1. List the 15 partitions of the set \(\{ \mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } \}\) in which A and E are in the same subset.
2. By considering the number of subsets in each of the partitions in part (a), or otherwise, explain why there are 8 partitions of the set \(\{ \mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } \}\) into two subsets with A and E in different subsets. Ali says that each of the 15 partitions from part (a) can be used to give two partitions in which A and E are in different subsets by moving E into a subset on its own or by moving E into another subset.
   [0pt]
3. 1. By considering the partition from part (a) into just one subset, show that Ali is wrong. [1]
   2. By considering a partition from part (a) into more than two subsets, show that Ali is wrong.

4

1. Show that there are 127 ways to partition a set of 8 distinct elements into two non-empty subsets. A group of 8 people ( \(\mathrm { A } , \mathrm { B } , \ldots\) ) have 8 reserved seats ( \(1,2 , \ldots\) ) on a coach. Seat 1 is reserved for person A , seat 2 for person B , and so on. The reserved seats are labelled but the individual people do not know which seat has been reserved for them. The first 4 people, \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D , choose their seats at random from the 8 reserved seats.
2. Determine how many different arrangements there are for the seats chosen by \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D . The group organiser moves \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D to their correct seats (A in seat \(1 , \mathrm {~B}\) in seat \(2 , \mathrm { C }\) in seat 3 and D in seat 4).
   The other 4 people ( \(\mathrm { E } , \mathrm { F } , \mathrm { G }\) and H ) then choose their seats at random from the remaining 4 reserved seats ( \(5,6,7\) and 8 ).
3. List the 9 derangements of \(\{ \mathrm { E } , \mathrm { F } , \mathrm { G } , \mathrm { H } \}\), where none of these four people is in the seat that has been reserved for them. Suppose, instead, that the 8 people had chosen their seats at random from the 8 reserved seats, without the organiser intervening.
4. Determine the total number of ways in which the seats can be chosen so that 4 of the people are in their correct seats and 4 are not in their correct seats.

1 A set consists of five distinct non-integer values, \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }\) and E . The set is partitioned into non-empty subsets and there are at least two subsets in each partition.

1. Show that there are 15 different partitions into two subsets.
2. Show that there are 25 different partitions into three subsets.
3. Calculate the total number of different partitions. The numbers 12, 24, 36, 48, 60, 72, 84 and 96 are marked on a number line. The number line is then cut into pieces by making cuts at \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }\) and E , where \(0 < \mathrm { A } < \mathrm { B } < \mathrm { C } < \mathrm { D } < \mathrm { E } < 100\).
4. Explain why there must be at least one piece with two or more of the numbers 12, 24, 36, 48, 60, 72, 84 and 96.

50 people are at a TV game show. 21 of the 50 are there to take part in the game show and the others are friends who are in the audience, 22 are women and 20 are from London, 2 are women from London who are there to take part in the game show and 15 are men who are not from London and are friends who are in the audience.

1. Deduce how many of the 50 people are in two of the categories 'there to take part in the game show', 'is a woman' and 'is from London', but are not in all three categories. [3]

The 21 people who are there to take part in the game show are moved to the stage where they are seated in two rows of seats with 20 seats in each row. Some of the seats are empty.

1. Show how the pigeonhole principle can be used to show that there must be at least one pair of these 21 people with no empty chair between them. [2]

The 21 people are split into three sets of 7. In each round of the game show, three of the people are chosen. The three people must all be from the same set of 7 but once two people have played in the same round they cannot play together in another round. For example, if A plays with B and C in round 1 then A cannot play with B or with C in any other round.

1. By first considering how many different rounds can be formed using the first set of seven people, deduce how many rounds there can be altogether. [3]