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Nilanjan Bhowmick AIR 3, CSIR NET (Earth Science)
- IIT JAM
- Mathematics (MA)
Can somebody explain image,preimage,one to one,onto with examples
TUTTGUTOT EIITEU :A mapping f: S> T is called one-to-one, if X yf(x) # f(y) Y X, y e S. Or f(x) = f(y) x = y VX, y e S.
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Srinath
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Consider a function f, between two non empty sets, f : S -> T. Then, the set of all elements f(x) which belongs to T, is called the image set of f. That is, the image set = { f(x) € T | x€S}. Note that, the image set is a subset of T. The elements of the image set are called images. Now, preimages are those x€ S, whose image is in T. Observe that, an image can have many preimages, but a preimage can have only one image( function definition). 1-1 function simply states, that distinct elements in S are mapped to distinct elements in T, (no two elements in S have the same image). Onto function, simply means, that the codomain, T is equal to the range, ( any element in T, will have a preimage in S).
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Thank you Sir. Now only I understood clearly
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Srinath Best Answer
Consider the function f : {1,2,3,4} -> {a,b,c,d}. Where f(1) = b, f(2) = a, f(3) = d, f(4) = c. Now a,b,c,d are all images, because they are all of the form, f(x) for some x in {1,2,3,4}. And the elements 1,2,3,4 are all preimages, because the image of them is in {a,b,c,d}. You can check whether this function satisfies the conditions of 1-1 and onto.
How to check whether it is onto or one to one
Use the conditions I have mentioned, in the comments. It'll be easy. If you find it difficult, I'll type it out for you. But you have to try.