Sets And Relations

Step 1: Understanding the function.
The given function is a cubic polynomial: which is a degree-3 polynomial. To determine if this function is one-one or onto, we need to analyze its properties.
Step 2: Checking if the function is one-one.
A function is said to be one-one (injective) if every element in the domain maps to a unique element in the codomain. In the case of a cubic polynomial like , since the polynomial has a degree of 3, it is not one-one because it will have repeated values for different -values due to the turning points in the graph. This can be confirmed by analyzing the derivative, which will show that the function has local maxima and minima.
Step 3: Checking if the function is onto.
A function is onto (surjective) if every element in the codomain has a pre-image in the domain. Since this is a cubic polynomial, as or , also tends to infinity or negative infinity. Hence, for every real value of , there exists a real value of such that , making the function onto.
Step 4: Conclusion.
Since the function is onto but not one-one, the correct answer is (b).

Step 1: Understanding the dual.
The dual of a Boolean expression is obtained by interchanging the AND ( ) and OR ( ) operations and replacing 1 by 0 and 0 by 1. Given the expression: to find its dual, we follow the rule for duality.
Step 2: Applying duality.
- Replace all (OR) with (AND). - Replace all (AND) with (OR). - Replace 1 with 0 and 0 with 1. The dual of the given expression is:
Step 3: Conclusion.
Thus, the correct dual is , and the correct answer is (a).

Step 1: Understanding the problem.
We are given two conditions: 1. , which means that the sum of and is 1 in Boolean algebra. 2. , which means that the product of and is 0 in Boolean algebra.
Step 2: Analyzing the first condition.
The equation implies that , because in Boolean algebra, if the sum of and is 1, one of them must be the complement of the other.
Step 3: Analyzing the second condition.
The equation means that the product of and is 0. Since , we know that and cannot both be 1 at the same time, making their product zero.
Step 4: Conclusion.
Thus, , and the correct answer is (d).