Find the sup and inf of each of the following setsSolutionFo

Find the sup and inf of each of the following sets:

Solution

For both a and b parts. The set is NATURAL NUMBERS which is countable infinity.

a.) the set becomes {1, 1/2, 1/3, 1/4........ 0}

Therefore, infimum is 0 and supremum is 1 for this set.

b.) the set becomes {1, 3/2, 5/3, 7/4, 9/5..... 2}

Therefore, infimum is 1 and supremum is 2 for this set.

c.) Here exponent is raised to the power of Rational number SET. So, taking a negative number (like -5000 or any higher number than this), we can find that the value of this function approaches 0.

Therefore, infimum for this set is 0.

On the other side, taking a large positive number, this function approaches +infinity

Thus, the supremum is +infinity for this set.

d.) As x can be equal to zero, thus the infimum for this set is 0.

Though x can not be equal to 2, but to find supremum, we will consider the nearest value, i.e. 4.

(it is a theoram of SET THEORY. which states that:

sup(x<b) = sup(x <= b) = b )