Relations And Functions

Step 1: We are given the relation , and we need to determine the properties of this relation.

Step 2: To check if the relation is reflexive, substitute into the equation: This is true for all values of , so the relation is reflexive.

Step 3: The relation is not symmetric or transitive, as demonstrated by further analysis, making the correct answer is only reflexive.

Step 1: The function is given, and we are asked to find a general pattern for , where for .

Step 2: First, calculate the first few terms:

• For , we have .
• For , we apply again to :

• For , apply again to :

Step 3: After observing the pattern for the first few terms, it becomes clear that:

Step 4: Now, substitute into the formula for each :

• For example:

Step 5: From the general pattern , it is clear that the product follows the form:

1. The function is defined for all .

2. To check if f is one-one: - Compute the derivative:

- Since for all and for , is strictly increasing for and strictly decreasing for . Therefore, is not one-one.

3. To check if f is onto: - The range of is:

- Since does not cover all real numbers ( ), is not onto.

4. Since is neither one-one nor onto, it is not bijective.

Step 1: First, determine the sets and :

• (even natural numbers less than 8)
• (prime integers less than 7)

Step 2: The number of relations from set to set is given by the total number of subsets of the Cartesian product . The number of elements in is:

Step 3: The number of relations is the number of subsets of , which is , since each pair in can either be included in or excluded from the relation. Thus, the correct answer is:

Step 1: The given function is:

. Let's first understand the behavior of the function by analyzing the terms involved.

Step 2: We need to examine whether the function is one-one (injective) and onto (surjective).

• A function is one-one (injective) if different inputs give different outputs, i.e., if implies .
• A function is onto (surjective) if for every possible output in the codomain , there exists an input such that .

Step 3: Analyze the function for injectivity:

• For to be injective, each output value must correspond to exactly one input. However, causes the function to behave identically for both positive and negative values of , thus violating injectivity. For example, , meaning the function is not one-one.

Step 4: Analyze the function for surjectivity:

• The function is limited in its range. It is bounded between and because both and grow exponentially, keeping the output within this interval. Therefore, it cannot cover all of , and hence the function is not onto.

Step 5: Therefore, the function is neither one-one nor onto.