Let A be the set of all 50 students of Class X in a school. Let f:A→N be function defined by f(x)= roll number of the student x. Show that f is one-one but not onto.
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Step-by-Step Solution
Step 1: Understand the Function Definition
We are given a set A containing 50 students from Class X. The function f maps each student in set A to their unique roll number, which is a natural number. The codomain of the function is the set of natural numbers, N={1,2,3,…}.
Step 2: Prove One-One (Injective)
To prove that a function is one-one, we assume that f(x1)=f(x2) for two elements x1,x2 in the domain A. If this assumption leads to x1=x2, then the function is one-one. In this case, if two students have the same roll number, they must be the same student, as roll numbers are unique identifiers.
Step 3: Prove Not Onto (Surjective)
For a function to be onto, every element in the codomain must have at least one pre-image in the domain. Here, the domain A has 50 elements, so the function can map to at most 50 distinct roll numbers. However, the codomain N (natural numbers) is infinite. This means there are infinitely many natural numbers that are not roll numbers of any student in Class X, and thus have no pre-image in A.
Step 4: Conclusion
We have successfully demonstrated that the function f is one-one because each student has a unique roll number. We have also shown that f is not onto because the set of natural numbers (codomain) contains elements that are not roll numbers of any student in the given class.