The integers less than or equal to 2 are: {2, 1, 0, -1, -2, -3, ...}
10, 8, 6, 4, 2, 0 and all the negative even integers.
Seven of them.
All integer less than or equal to 3 would actually be infinite. It would start at 3 and would keep decreasing in value. 3, 2, 1, 0, -1, -2, and so on forever.
3, 2, 1
6
That happens when only one of the two integers is negative.
10, 8, 6, 4, 2, 0 and all the negative even integers.
Seven of them.
To find how many positive integers less than or equal to 10,000 are divisible by 2, 5, or 10, we can use the principle of inclusion-exclusion. The count of integers divisible by each is: Divisible by 2: ( \frac{10000}{2} = 5000 ) Divisible by 5: ( \frac{10000}{5} = 2000 ) Divisible by 10: ( \frac{10000}{10} = 1000 ) Applying inclusion-exclusion, we have: [ 5000 + 2000 - 1000 = 6000 ] Thus, there are 6000 positive integers less than or equal to 10,000 that are divisible by 2, 5, or 10.
The integers that are greater than -2 but less than 5 are: -1, 0, 1, 2, 3, 4
The odd integers less than 5 are 1 and 3. Therefore, there are 2 odd integers that meet this criterion.
All integer less than or equal to 3 would actually be infinite. It would start at 3 and would keep decreasing in value. 3, 2, 1, 0, -1, -2, and so on forever.
17
1 and 2
There are no negative integers greater than 2. Negative integers are less than zero, while the integer 2 is positive. Therefore, the set of negative integers consists of numbers like -1, -2, -3, and so on, which do not exceed 2.
-2 is less than 2
A counterexample to the statement "the difference of two integers is less than either integer" can be demonstrated with the integers 5 and 3. The difference is (5 - 3 = 2). Here, 2 is not less than either integer, as it is less than 5 but greater than 3. Thus, this example shows that the difference can be less than one integer but not the other.