Small Hint: the sets that you are asked to prove something about are open sets, and open sets can be written as the countable union of closed sets i. Each of those sets can then be viewed as part of a sequence of functions that you can take a limit of.
For general intuition: this is something that tends to come with time. I found I needed to think a lot about these kinds of things before I had a library of tools that let me look at something and know where to start. Hunting for counter-examples tends to help a great deal as it forces you to really understand what you're trying to work with, and finding them can help direct you towards the right route when you get stuck on other problems.
Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Asked 5 months ago. Active 5 months ago. Viewed 57 times. The set containing no elements at all is called the null set , or empty set.
It may seem odd to define a set that contains no elements. Bear in mind, however, that one may be looking for solutions to a problem where it isn't clear at the outset whether or not such solutions even exist. If it turns out that there isn't a solution, then the set of solutions is empty.
Operations performed on the empty set as a set of things to be operated upon can also be confusing.
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Such operations are nullary operations. For example, the sum of the elements of the empty set is zero, but the product of the elements of the empty set is one see empty product. Ultimately, the results of these operations say more about the operation in question than about the empty set. For instance, notice that zero is the identity element for addition, and one is the identity element for multiplication.
The 'counting' numbers or whole numbers starting at 1, are called the natural numbers. This set is sometimes denoted by N. All whole numbers, positive, negative and zero form the set of integers.
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It is sometimes denoted by Z. If we expand the set of integers to include all decimal numbers, we form the set of real numbers.
The set of reals is sometimes denoted by R. A real number may have a finite number of digits after the decimal point e. In the case of an infinite number of digits, these digits may:.
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Those real numbers whose decimal digits are finite in number, or which recur, are called rational numbers. The set of rationals is sometimes denoted by the letter Q. If q equals 1, the fraction is just the integer p. Note that q may NOT equal zero as the value is then undefined.
Click the link for Set Theory Exercise 1. In this case, we simply write:. You may recall that when we defined the empty set we noted that there may be no solutions to a particular problem - hence the need for an empty set. Well, here we may be trying several different approaches to solving a problem, some of which in fact lead us to the same solution.
When we come to consider the distinct solutions, however, any such repetitions would be ignored. In such a case we would write:. Notice also that every set is a subset of the universal set , and the empty set is a subset of every set.
In other words:. On the left, the sets A and B are disjoint , because the loops don't overlap. On the right A is a subset of B , because the loop representing set A is entirely enclosed by loop B. Note that the rectangle representing the universal set is divided into four regions, labelled i , ii , iii and iv.
So A is a subset of B , and the diagram should be re-drawn like Fig 2 above.
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