When finding the support of the closed item set. What if there are two smallest closed item sets for lemon such as {Pasta, lemon} support 3 and {Orange, Lemon} support 4. What is the support of lemon in this case? is it 3 or 4?
Hi thanks for watching! It is a good question and it was fun to think about it to answer you. The answer is that this situation is impossible. Let me explain. Here I will use L = Lemon, P= Pasta and O = Orange to explain briefly. Let T(OL) be the set of transactions containing OL Let T(PL) be the set of transactions containing PL Let T(L) be the set of transactions containing L The assumption in your question is that OL is closed, PL is closed and L is not closed. I will show it is impossible. If we assume that OL and PL are both closed, then T(OL) and T(PL) must be disjoint. That means that there exists at least some transaction where OL and PL do not appear together. Why? Because if T(OL) and T(PL) would be the same, then OPL would be closed and OL and PL would not be closed. So we must rule out this possibility. So, based on this, it means that T(OL) and T(PL) must be disjoint. But if T(OL) and T(PL) are disjoint, it means that the support of L must be greater than OL and PL. Why? Because T(L) = T(OL) U T(PL) and T(OL) and T(PL) would have to be disjoint. So in your question, if support(PL) = 3 and support(OL) = 4, the support of Lemon must be at least 5. If the support of Lemon is 5. It means that Lemon is closed, which contradict the assumption of the question. But what is Lemon was not closed? Then it would mean that there exist another item like Sugar such that T(Sugar,Lemon) = T(L) but this is impossible because in that case OL and PL would not be closed. So this situation will not happen. This is not a formal proof. But I think you can see that it is impossible 🙂
Thank you professor, very helpful explanation!
I have been looking for a good explanation of these topics. Thank you so much for making these videos. u explain like a god
Thanks for watching and for your nice feedback!
thank you for your detailed and clear explanation. it was very helpful and understandable.
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When finding the support of the closed item set. What if there are two smallest closed item sets for lemon such as {Pasta, lemon} support 3 and {Orange, Lemon} support 4. What is the support of lemon in this case? is it 3 or 4?
Hi thanks for watching! It is a good question and it was fun to think about it to answer you. The answer is that this situation is impossible.
Let me explain. Here I will use L = Lemon, P= Pasta and O = Orange to explain briefly.
Let T(OL) be the set of transactions containing OL
Let T(PL) be the set of transactions containing PL
Let T(L) be the set of transactions containing L
The assumption in your question is that OL is closed, PL is closed and L is not closed. I will show it is impossible.
If we assume that OL and PL are both closed, then T(OL) and T(PL) must be disjoint. That means that there exists at least some transaction where OL and PL do not appear together. Why? Because if T(OL) and T(PL) would be the same, then OPL would be closed and OL and PL would not be closed. So we must rule out this possibility.
So, based on this, it means that T(OL) and T(PL) must be disjoint.
But if T(OL) and T(PL) are disjoint, it means that the support of L must be greater than OL and PL. Why? Because T(L) = T(OL) U T(PL) and T(OL) and T(PL) would have to be disjoint.
So in your question, if support(PL) = 3 and support(OL) = 4, the support of Lemon must be at least 5.
If the support of Lemon is 5. It means that Lemon is closed, which contradict the assumption of the question.
But what is Lemon was not closed? Then it would mean that there exist another item like Sugar such that T(Sugar,Lemon) = T(L) but this is impossible because in that case OL and PL would not be closed.
So this situation will not happen. This is not a formal proof. But I think you can see that it is impossible 🙂
thank you so much!
Thanks. Glad you like it!