Functions

A function is said to be onto (or surjective) if for every element , there exists an element such that . This means that the image of under the function must cover the entire set . Hence, for to be onto, the condition is: This means the range of the function is exactly equal to , which corresponds to option (B).

We are given that , so we first need to find . Using the given operation: Now, we need to find : Thus, the correct answer is .

The cardinality of the Cartesian product is given by: Given and , we can substitute these values: Thus, the correct answer is .

For a function to be onto (surjective), it must map every element in the domain to an element in the co-domain such that the range of covers all of . This means that the image of , denoted by , must be equal to the entire set . Thus, for to be an onto function: Therefore, the correct answer is: (B) .

We are given the function . To find the inverse , we will follow these steps: 1. Replace with : 2. Solve for in terms of : 3. Now, replace with to get the inverse function: Thus, the inverse function is .

We are given the operation . First, calculate : Next, calculate using the same operation: So, . Thus, the correct answer is 9. So, the answer is (C) 9. There was a misunderstanding in the previous evaluation.

For a function to be bijective, it must be both injective (one-to-one) and surjective (onto). This means that every element in the domain must map to a unique and distinct element in the codomain . The number of ways to define a bijective function from a set of elements to itself is given by the number of permutations of the set, which is . Thus, the correct answer is .