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A bijection is a one-to-one correspondence in set theory - a function which is both a surjection and an injection.
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Which set of real numbers is bigger set of integers or set of rational numbers explain?
The question is not well-posed, in that the term "bigger" can be understood in different ways. If A is a subset of B, we can call B bigger than A. However, in set theory, the cardinality of a set is defined as the class of sets with the "same number" of elements: Two sets A and B have the same cardinality if there exists a bijection f:A->B. Theorem: If there is an injection i:A->B and an injection i:B->A, then there is a bijection f:A->B. Not proved here. The set of integers and the set of rational numbers can be mapped as follows. Since the natural numbers are a subset of the rational numbers by i:N->R: n-> n/1, we have half of the proof. Now, order the rational numbers as follows: - assign to each rational number p/q (p,q > 0) the point (p,q) in the plane. Next, consider that you can assign a natural number to each rational number by walking through them in diagonals: (1,1) -> 1; (2,1) -> 2; (1,2) -> 3; (3,1) ->4 ; (2,2) ->5; (1,3) -> 6; (4,1) -> 7; (3,2) -> 8, (2,3) -> 9; (1,4) -> 10, etc. (make a drawing). It is clear that in this way you can assign a unique natural number to EACH rational number. This means that you have an injection from the rational numbers to the natural numbers. Now you have two injections, from the natural numbers to the rational numbers and from the rational numbers to the natural numbers. By the theorem, there is a bijection, which means that the natural numbers and the rational numbers have the same cardinality. Neither of them is "bigger" than the other in this sense. The cardinality of these two sets is called Aleph-zero, and the sets are also called countable (because the elements can be counted with the natural numbers).
Is the inverse of f of x a function?
If f(x) is a function, the inverse may, or may not, be a function. In math, quite often it is possible, and sensible, to restrict the original function to a certain range of numbers, within which the inverse is well-defined.The function f(x) has an inverse (within a certain range) if it is strictly monotonous within that range.
What is a bijection?
A bijection is a one-to-one correspondence in set theory - a function which is both a surjection and an injection.
Does the greatest integer function have an inverse function?
Inverse of a function exists only if it is a Bijection. Bijection=Injection(one to one)+surjection (onto) function.
What are some examples of Injection Surjection and Bijection?
Too long to explain so just go here http://en.wikipedia.org/wiki/Bijection,_injection_and_surjection
Is a monotonic function a bijection?
No. For example, y = 7 is monotonic. It may be a degenerate case, but that does not disallow it. It is not a bijection unless the domain and range are sets with cardinality 1. Even a function that is strictly monotonic need not be a bijection. For example, y = sqrt(x) is strictly monotonic [increasing] for all non-negative x. But it is not a bijection from the set of real numbers to the set of real numbers because it is not defined for negative x.
When does a relation be a function?
A function is a relation whose mapping is a bijection.
What is the difference between relations and functions?
A function is a relation whose mapping is a bijection.
What is the correspondence of the domain and range of a linear function?
It is a bijection [one-to-one and onto].
Is every bijection a strictly monotonic function?
No. For example, consider the discontinuous bijection that increases linearly from [0,0] to [1,1], decreases linearly from (1,2) to (2,1), increases linearly from [2,2] to [3,3], decreases linearly from (3,4) to (4,3), etc.
What is a bijective numeration?
A bijective numeration is a numeral system which uses digits to establish a bijection between finite strings and the positive integers.
What words have two J's in it?
Jigging jigs, and jujitsu are some. Answer Jejunum
Is every permutation always a one-to-one function?
By definition, a permutation is a bijection from a set to itself. Since a permutation is bijective, it is one-to-one.
Why does multiplication by 2 define a bijection from R to R R meaning set of Real numbers but not from Z to Z Z meaning set of integers?
It's not a bijection in Z because it's not surjective. For example, f(x) = 3 has no solution in Z. In other words, you can't double an integer (Z) to get an odd number. It works in R because it's ok to have decimals.
