Question

Find the infimum and the supremum, if they exist, of each of the following sets:

a: { x in R : 2x - 5 > 0 }
b: { x in R : x² >= x² }
c: { x in R : x < 1/x }
d: { x in R : x² - 2x - 5 < 0 }
a) (-[infinity], [infinity]); (-[infinity], [infinity]); Not exist; (-[infinity], 5)
b) (-[infinity], [infinity]); (-[infinity], [infinity]); Not exist; (-[infinity], 5)
c) (5/2, [infinity]); Not exist; (-[infinity], 0) and (1, [infinity]); (-[infinity], 1) and (3, [infinity])
d) (5/2, [infinity]); Not exist; (-[infinity], 0) and (1, [infinity]); (-[infinity], 1) and (2, [infinity])

Answer 1

The infimum and supremum of each set in the question

Step-by-step explanation:

a) The set { x in R : 2x - 5 > 0 } represents all real numbers that are greater than 5/2. The infimum of this set is 5/2 because it is the smallest value that satisfies the inequality. The supremum does not exist because there is no largest value in the set.

b) The set { x in R : x² >= x² } represents all real numbers. Both the infimum and supremum of this set are -∞ and +∞, respectively, because there is no lower or upper bound.

c) The set { x in R : x < 1/x } represents all real numbers greater than 0 and less than 1. The infimum of this set is 0 because it is the smallest value that satisfies the inequality. The supremum does not exist because there is no largest value in the set.

d) The set { x in R : x² - 2x - 5 < 0 } represents all real numbers between 1 and 3. The infimum of this set is 1 because it is the smallest value that satisfies the inequality. The supremum is 3 because it is the largest value in the set.