Question

The integers greater than 10. (Check all that apply.) Check All That Apply

a. The set is countably infinite.
b. The set is finite.
c. The set is countably infinite with one-to-one correspondence 1艹11, 2艹12, 3-13, and so on.
d. The one-to-one correspondence is given by n(n+10)
e. The set is countably infinite with one-to-one correspondence 1艹10, 2·-12.3艹17, and so on.
f. The one-to-one correspondence is given by n艹(2n+10).

Answer 1

The set of integers greater than 10 is countably infinite and can be put into a one-to-one correspondence with the natural numbers using the formula n ↔ n+10. Statements regarding it being finite or correspondences using incorrect formulas are incorrect.

Step-by-step explanation:

The question is asking about the characteristics of the set of integers greater than 10. To evaluate the given statements:

• The set of integers greater than 10 is countably infinite. This means that the set has an infinite number of elements but can be put into a one-to-one correspondence with the natural numbers.
• Statement a is correct because you can list the integers greater than 10 in a sequential order starting from 11, 12, 13, and so on, which demonstrates its countably infinite nature.
• Statement b is incorrect as the set of integers greater than 10 is not finite; there is no upper limit to the numbers.
• Statement c and e are incorrect because they provide various incorrect ways of creating a correspondence. The correct correspondence would be simply n ↔ n+10.
• Statement d is incorrect because n(n+10) does not create the integers greater than 10; it produces a different sequence of numbers.
• Statement f provides an incorrect formula for one-to-one correspondence. The correct formula for mapping a counting number n to an integer greater than 10 would be n ↔ n+10.