Question

$A\to B$ will be an into function if
A. $f\left( A \right)\subset B$
B. $f\left( A \right)=B$
C. $B\subset f\left( A \right)$
D. $f\left( B \right)\subset A$

Answer 1

Hint: As the condition given in the question states that the function is an into a function, so we have to think about what an into function means and about how the elements that are related to each other.

Complete step by step answer:
In the question, an expression of $A\to B$ is given and we have been asked to find the correct option for which f is an into function. Before finding the correct answer, let us find out about what a function is. IN mathematics, a function is a binary relation between two sets that associates every element of the first set to exactly one element of the second set. The concept of a function was formalized at the end of the ${{19}^{th}}$ century in terms of set theory and this greatly enlarged the domains of the application of the concept. The typical examples of this are functions from integers to integers or from real numbers to real numbers. Functions were originally the idea of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically this concept was elaborated with infinitesimal calculus at the end of the ${{17}^{th}}$ century, that is the functions were considered as differentiable. (That is they had a high degree of regularity) It is customarily denoted by letters such as $f,g,h$.
Now, in the question we are given that $f$ is an into the function, which means that if there is a function, let’s say from A to B, then at least set B has an element which is not connected with any of the elements of set A. So, that means we can say that all the elements or values of $f\left( A \right)$ will be included in B. We can also write this as $f\left( A \right)\subset B$. Therefore, the correct answer is option A.

Note:
The students generally get confused by ‘into function’ and ‘onto function’. The students should remember that in and ‘into function’ there would be an extra element in the range set, whereas in ‘onto function’ there would not be any extra element and every element would be related.