In this question, I wish to ask while calculating the mean of S if we would consider element b twice or not. According to me, in sets identical elements are allowed but while counting the number of elements, we would not count duplicates right. So, would the mean be (a+b+c)/3 or (a+2b+c)/4. My doubt is not necessarily regarding how to solve this QC in general but more on this very specific set concept. Would be grateful for the answer.
When something is explicitly defined as a set, duplicate elements are neither included in the cardinality nor considered when calculating the mean. So the mean of S here would be (a+b+c)/3.
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Set A = \{a,b,b,c\} is identical to Set B =\{a,b,c\} because B \subseteq A and A \subseteq B, hence A = B. This implies that they both have cardinality (size) of 3.
Once we’ve established that, the answer to your other questions follow naturally too.
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Thank you both for your replies. I recently came across the concept of Multiset/List Behavior, which includes duplicates and counts each occurrence of every element. Since this seems similar to a set, how do we differentiate between the two? If the question specifically mentions a duplicate, shouldn’t we consider it a Multiset? I’m eager to learn more about this concept of multiset as this is new to me. I would appreciate any clarification you can provide. Thank you!
It depends on context and there are other notations to denote a multiset as well. It’s also good practice for the author to tell you straightaway that you’re dealing with a multiset.
Nonetheless, you don’t encounter these on the gre or even on any intro level texts cuz they’re not broadly useful other than to generalize the notion of a set.
I see. Thank you so much for clarifying.