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mruncleramos
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Can a mapping from f:S->T associate an element of s into several elements of T? Also, how do you prove: A mapping f:S->T is bijective if and only if it has an inverse?
mruncleramos said:Can a mapping from f:S->T associate an element of s into several elements of T? Also, how do you prove: A mapping f:S->T is bijective if and only if it has an inverse?
A mapping, also known as a function, is a relation between two sets where each element in the first set (domain) is paired with exactly one element in the second set (codomain). The mapping is denoted as f: S->T, where S is the domain and T is the codomain.
A bijective mapping is one that is both injective (one-to-one) and surjective (onto). This means that each element in the codomain is paired with exactly one element in the domain, and each element in the codomain has at least one corresponding element in the domain. In other words, every element in the codomain is mapped to by exactly one element in the domain, and every element in the codomain is mapped from by exactly one element in the domain.
To prove that a mapping is bijective, you must show that it is both injective and surjective. To prove injectivity, you must show that for every pair of distinct elements in the domain, their corresponding elements in the codomain are also distinct. To prove surjectivity, you must show that every element in the codomain has at least one corresponding element in the domain.
An inverse mapping, denoted as f-1, is a mapping that undoes the effects of the original mapping f. In other words, if f maps an element x in the domain to an element y in the codomain, f-1 maps y back to x. This means that f-1 is the reverse of f.
A bijective mapping must have an inverse in order to satisfy the definition of a bijective mapping. If a mapping is not bijective, it may not have an inverse, as there may not be a one-to-one correspondence between the elements of the domain and the elements of the codomain. Therefore, the existence of an inverse is a necessary condition for a mapping to be considered bijective.