Sets

Step 1: Understand the definition of a null set.
A null set, or empty set, is a set that contains no elements.
It is denoted by or .
Step 2: Analyze the options.
Option (a): : This set is empty because there is no value of that can satisfy both conditions simultaneously (a number cannot be both greater than 1 and less than 1 at the same time).
Therefore, , which is a null set.
Option (b): : Solving gives , so this set contains the element 0.
Hence, it is not a null set.
Option (c): : This is explicitly the null set, as it is empty by definition.
Option (d): : This set contains all values of that are equal to 1.
Hence, it is not empty; it contains the element 1.
Step 3: Conclusion.
Option (a) is the null set, as no value of can satisfy the condition simultaneously.

Step 1: Understand the notation. The symbol means "is an element of".
So, when we say , we are stating that 3 is an element of the set .
Step 2: Analyze the options.
Option (a): : This is true because 3 is an element of the set .
Option (b): : This is identical to option (a) and is also true.
Option (c): : This is false because is a set containing 3, not the number 3 itself.
The set is not an element of ; the number 3 is.
Option (d): : This is false because is a set, and is not an element of the set .
The elements of the set are just 1, 3, and 5, not the set .
Step 3: Conclusion.
The correct statement is option (a), which states that 3 is an element of the set .