- #1
fiksx
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- Homework Statement
- If the proposition below is true, show this. If false, give a counterexample.
From the integers from 1 to 11, make 10 sets ##S_1,S_2, \dots, S_{10}## each with 4 integers selected. Within each sets, the same number shall not be chosen twice. No matter how ##S_1,S_2, \dots, S_{10}## is selected, two sets ##S_i,S_j## (i not equal to j) include two or more common integers.
- Relevant Equations
- Sets
Is this related to pigeon principle?
$$S_1=\{1,2,3,4\},$$
$$S_2=\{2,3,4,5\},$$
$$S_3=\{4,5,6,7\},$$
$$S_4=\{5,6,7,8\},$$
$$S_5=\{7,8,9,10\},$$
$$S_6=\{8,9,10,11\},$$
$$S_7=\{5,6,2,4\},$$
$$S_8=\{1,5,7,9\},$$
$$S_9=\{4,8,10,11\},$$
$$S_{10}=\{5,7,10,11\}$$
When we choose two of them, there is possibility there are same integer but not all ?
If this is related to pigeonhole principle or inclusion exclusion principle, Is there possibility that sets are hole and number 1-11 is pigeon but what is the relation with in each sets there are four number?
$$S_1=\{1,2,3,4\},$$
$$S_2=\{2,3,4,5\},$$
$$S_3=\{4,5,6,7\},$$
$$S_4=\{5,6,7,8\},$$
$$S_5=\{7,8,9,10\},$$
$$S_6=\{8,9,10,11\},$$
$$S_7=\{5,6,2,4\},$$
$$S_8=\{1,5,7,9\},$$
$$S_9=\{4,8,10,11\},$$
$$S_{10}=\{5,7,10,11\}$$
When we choose two of them, there is possibility there are same integer but not all ?
If this is related to pigeonhole principle or inclusion exclusion principle, Is there possibility that sets are hole and number 1-11 is pigeon but what is the relation with in each sets there are four number?