State whether each of the following set is finite or infinite:
The set of lines which are parallel to the x-axis
Infinite: There are infinite lines parallel to the x-axis. Since the number of elements is not definite, the set is infinite.
The set of letters in the English alphabet
Finite: The set of letters in the English alphabet has a definite number of elements (26). A set consisting of a definite number of elements is finite.
The set of numbers which are multiple of 5
Infinite: The set of numbers which are multiples of 5 is {5, 10, 15, ...}. Like the set of natural numbers, it does not have a definite number of elements, so it is infinite.
The set of animals living on the earth
Finite: The number of animals living on the earth is a definite natural number. Similar to the set of men living presently mentioned in the text, this set is finite.
The set of circles passing through the origin (0,0)
Infinite: There are infinite circles that can pass through the origin (0,0). Since the number of such circles is not definite, the set is infinite.
Explanation
The answers are derived from Definition 2 in the text, which states a set is finite if it is empty or has a definite number of elements; otherwise, it is infinite. For part (iv), the text explicitly compares similar sets (men living presently) to finite sets where the number of elements is a natural number. For parts (i), (iii), and (v), the reasoning follows the examples of natural numbers and points on a line provided in the text, where elements do not follow a definite ending, making the sets infinite.