Can anyone please explain why the one is not the correct answer?
Let P(n,m) be a property about two integers n and m. If we want to disprove the claim that "For every integer n, there exists an integer m such that P(n,m) is true", then we need to prove that
1)There exists an integer n such that P(n,m) is false for all integers m.
2)For every integer m, there exists an integer n such that P(n,m) is false.
3)There exists integers n,m such that P(n,m) is false.
4)There exists an integer m such that P(n,m) is false for all integers n.
5)For every integer n, and every integer m, the property P(n,m) is false.
6)For every integer n, there exists an integer m such that P(n,m) is false.
7)If P(n,m) is true, then n and m are not integers.
Why the 3rd one is not the correct answer and what is the correct answer?
To explain why option 1) is not the correct answer, let's break down the logic behind each option:
1) There exists an integer n such that P(n,m) is false for all integers m.
This option is asking for an n such that no matter the value of m, P(n,m) is always false. In other words, there is no m for which P(n,m) is true. This does not disprove the claim because it is possible for other values of n to have a true P(n,m) statement.
2) For every integer m, there exists an integer n such that P(n,m) is false.
This option is asking for an n such that for every value of m, P(n,m) is false. In other words, for any chosen m, there is an n that makes P(n,m) false. This also does not disprove the claim because it is focused on finding an n for each m, but doesn't cover all possible n.
3) There exists integers n, m such that P(n,m) is false.
This option is asking for the existence of at least one pair of integers n and m for which P(n,m) is false. It covers the possibility that there might be specific values of n and m that contradict the claim, but it does not ensure that for every value of n, there is an m making P(n,m) false.
4) There exists an integer m such that P(n,m) is false for all integers n.
This option is asking for an m such that for all values of n, P(n,m) is false. It is similar to option 1) but focused on finding an m for each n. Just like option 1), it does not disprove the claim since it is possible for other values of m to have a true P(n,m) statement.
5) For every integer n, and every integer m, the property P(n,m) is false.
This option says that for every possible combination of integer values for n and m, P(n,m) is always false. It covers all possible scenarios, but it doesn't disprove the claim. It is stating that the property P(n,m) is universally false, but the claim is about the existence of at least one true statement.
6) For every integer n, there exists an integer m such that P(n,m) is false.
This option is asking to find an m for each value of n such that P(n,m) is false. It is similar to option 2), but it does not cover the possibility that some values of n might have a true P(n,m) statement.
7) If P(n,m) is true, then n and m are not integers.
This option introduces a condition that if P(n,m) is true, then n and m cannot be integers. While this helps narrow down the potential solutions, it does not prove/disprove the claim since it doesn't consider the possibility of P(n,m) being false for some pairs of integers n and m.
Based on the explanations above, the correct answer to disprove the claim is option 3) - There exists integers n, m such that P(n,m) is false.