earlier, i used to think that i know very well about these terms in sets but.... omg... they were not so as they used to apear... they are so so so different from what i used to think of them...
please give me materials relating to them in order that i can understand them.. i know about functions.
please give me materials relating to them in order that i can understand them.. i know about functions.
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Well, a set is finite if it has a, well, finite number of things in them. For example, {1, 2, 3} has three things in it, making it a finite set. An infinite set is a set that is not finite, which has no finite limit to the number of things in it. For example, all integers {..., -3, -2, -1, 0, 1, 2, 3, ...}. Another way to think about it is, if you keep removing things, one by one, from a finite set, you'll empty the set after a while, but if you keep removing things from an infinite set, you'll never stop.
Now, out of the infinite sets, we can discern countable and uncountable sets. These are two types of infinite sets, where the uncountable ones are considered larger than the countable ones. The definition of a countable set is a set S for which there exists a function f : S ---> N (where N is the natural numbers) which is a bijection (i.e. both one-to-one and onto). Intuitively, a set is countable if you can arrange all of its elements into an infinite list. For example, the integers are countable, because we can arrange them like so:
0, 1, -1, 2, -2, 3, -3, 4, -4, ...
which will cover every integer exactly once. The bijection with N comes from counting as you read off the numbers (so 0 ---> 1, 1 ---> 2, -1 ---> 3, 2 ---> 4, -2 ---> 5, etc). Even the rational numbers are countable (but producing a bijection is a pain).
Uncountable sets are sets that are not countable. Such sets include, the real numbers, the complex numbers (which contains the real numbers), the set of subsets of the natural numbers, the set of integer-valued sequences, the set of continuous real functions, etc.
Now, out of the infinite sets, we can discern countable and uncountable sets. These are two types of infinite sets, where the uncountable ones are considered larger than the countable ones. The definition of a countable set is a set S for which there exists a function f : S ---> N (where N is the natural numbers) which is a bijection (i.e. both one-to-one and onto). Intuitively, a set is countable if you can arrange all of its elements into an infinite list. For example, the integers are countable, because we can arrange them like so:
0, 1, -1, 2, -2, 3, -3, 4, -4, ...
which will cover every integer exactly once. The bijection with N comes from counting as you read off the numbers (so 0 ---> 1, 1 ---> 2, -1 ---> 3, 2 ---> 4, -2 ---> 5, etc). Even the rational numbers are countable (but producing a bijection is a pain).
Uncountable sets are sets that are not countable. Such sets include, the real numbers, the complex numbers (which contains the real numbers), the set of subsets of the natural numbers, the set of integer-valued sequences, the set of continuous real functions, etc.
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{1,2,3--- } infinite set {1,2,3} finite set {days of a week} finite set set N,Q,R are infinite sets