You are right there !umzung said:
To say that "S" is a subset of "T" in R³ means that all the elements in "S" are also elements of "T", and that "S" is a smaller set than "T". In other words, every point in "S" can also be found in "T" and "S" does not contain any additional elements that are not in "T".
To prove that "S" is a subset of "T" in R³, you need to show that all the elements in "S" are also elements of "T". This can be done by taking a point from "S" and showing that it is also in "T". Additionally, you can show that "S" does not contain any elements that are not in "T". This can be done by taking a point that is not in "T" and showing that it is also not in "S".
There are a few common methods used to prove that "S" is a subset of "T" in R³. One method is to use direct proof, where you take an element from "S" and show that it is also in "T". Another method is to use a proof by contradiction, where you assume that there is an element in "S" that is not in "T" and show that this leads to a contradiction. You can also use a proof by contrapositive, where you show that if an element is not in "T", then it is also not in "S".
Yes, "S" and "T" can be equal sets if "S" is a subset of "T" in R³. This means that all the elements in "S" are also in "T" and there are no additional elements in "T". In this case, "S" is technically a subset of "T", but it is also equal in size and contains all the same elements as "T".
Yes, it is possible for "S" to be a proper subset of "T" in R³. This means that "S" is a subset of "T", but "S" is smaller than "T" and there are elements in "T" that are not in "S". In this case, "S" is a proper subset of "T" and "T" is a proper superset of "S".