# There exists a real number such that an integer is greater than it, right?

• Feb 8th 2011, 10:39 AM
Lord Darkin
There exists a real number such that an integer is greater than it, right?
Hello,

-If there's a value "a" such that it's a real number, and there exists a "b" such that it's an integer and that b > a
-If there exists an "a" such that it's a real number and that, for all of "b" that's a real number, b> a

Both of these statements should be true, right? Because no matter what you can always add 1 more to any numbers. But apparently the first statement I listed is true, but the second is false.
• Feb 8th 2011, 10:42 AM
dwsmith
Quote:

Originally Posted by Lord Darkin
Hello,

-If there's a value "a" such that it's a real number, and there exists a "b" such that it's an integer and that b > a
-If there exists an "a" such that it's a real number and that, for all of "b" that's a real number, b> a

Both of these statements should be true, right? Because no matter what you can always add 1 more to any numbers. But apparently the first statement I listed is true, but the second is false.

The first statement says there exist a "b" that is > than a.

The second statement says all b are > than a.

If our set is the reals, the first is true and the second is false.
• Feb 8th 2011, 10:51 AM
Lord Darkin
Thanks!
• Feb 8th 2011, 11:04 AM
emakarov
Quote:

If there's a value "a" such that it's a real number, and there exists a "b" such that it's an integer and that b > a
I agree with dwsmith's post above, but, to be strict, the quoted statement is not well-formed from the standpoint of either mathematics or English. A sentence that has an "if" must also have a "then," even if it is only implied. The correct statement is, "If $a$ is a real number, then there exists an integer b such that b > a."