Relations And Functions

Given,
and set of integers.

Range of set

is one value of x gives two values of y. So,
is not a function.

Step 1: Understanding the Relation
The relation is defined by the equation .
The domain of the relation consists of all values of for which there exists a corresponding value of in (natural numbers) that satisfies the relation.
Step 2: Solving the Equation for Possible and
We can solve for different values of and :
For , (valid)
For , (valid)
For , (valid)
For , (valid)
Thus, the domain of is .
Step 3: Conclusion
The domain of is .

Step 1: Understanding the Relation
The notation means that the relation is equal to its inverse.
In other words, for every pair in , the pair must also be in .
This property defines a symmetric relation.
Step 2: Explanation of Other Options
Option (a) Reflexive is incorrect because reflexivity means that must be in for all elements in , which is not necessarily true for .
Option (c) Transitive is incorrect because transitivity means if and , then must also be in , which is not implied by .
Option (d) None of these is incorrect because the relation is symmetric.
Step 3: Conclusion Thus, the relation is symmetric.

Step 1: Understand the condition for the inverse function.
For a function to have an inverse, it must be both one-to-one (injective) and onto (surjective).
A one-to-one function ensures that every element of the domain maps to a unique element in the codomain, and an onto function ensures that every element in the codomain has a corresponding element in the domain.
Step 2: Analyze the options.
Option (a): is one-one: A one-to-one function is necessary for the existence of the inverse, but it is not sufficient alone.
The function also needs to be onto. Option (b): is onto: Being onto is necessary but not sufficient.
A function must be both one-to-one and onto for an inverse to exist. Option (c): is one-one but not onto: This is not enough for the inverse to exist because the function is not onto. Option (d): is one-one and onto: This is the correct condition.
A function that is both one-to-one and onto is bijective, and only bijective functions have inverses.
Step 3: Conclusion.
For an inverse function to exist, the function must be bijective (both one-to-one and onto).