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# Logic and counting problems

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1. Each of the following people is either lying or telling the truth:

Aisha: If Bian is telling the truth, then Chang is lying.
Bian: If Chang is telling the truth, then so is Diego.
Chang: If Diego is lying, then Aisha is lying too.
Diego: Aisha is lying.

Determine all possible combinations of which people are lying and which are telling the truth. Back up your conclusion with a truth table or with other reasoning.

2. How many ways are there for 8 students to work in four clusters? Justify your reasoning. A cluster has to have at least one person in it. If the students are labeled A, B, C, D, E, F, G, H then some possible different sets of clusters are {ABC, DE, FH, G}, {ABC, DF, EH, G}, {A, B, C, DEFGH} and {EH, AG, BD, CF}. The order within each cluster and between clusters doesn't matter - ABC working together is the same as BAC working together and {EH, AG, BD, CF} = {BD, CF, EH, AG}.

Extension: How many ways are there for n students to work in 2 clusters? In 3 clusters? In 4 clusters?