Bachelier said:How do I prove this though. the part about the singleton
Dick said:If a space has only one point, it's clearly countable isn't it? Forming the CP isn't the challenge, it's showing a metric space with two points in it that's pathwise connected is uncountable. Any ideas on that one?
micromass said:You seemed to have changed "connected" into 'pathwise" connected. The OP was talking about just connected. The statement is true in both cases though (as I'm 100% sure you know), but it's much easier to prove if you take pathwise connected.
Bachelier said:I am trying to write a CP for:
Every connected M. Space with at least 2 points is uncountable.
Restatement:
if a MS X is connected with |X|≥ 2 => X is uncountable.
Contrapositive:
a MS X has only one point => X is not connected.
Thanks
Bachelier said:Restatement:
if a MS X is connected with |X|≥ 2 => X is uncountable.
Contrapositive:
a MS X has only one point => X is not connected.
A contrapositive statement is a logical statement that is formed by switching the hypothesis and conclusion of a conditional statement and negating both. For example, the contrapositive of "If it rains, then the ground is wet" is "If the ground is not wet, then it did not rain."
Checking for the contrapositive of a statement is important because it allows you to determine the validity of a conditional statement. If the original statement is true, then the contrapositive statement must also be true. This is a useful tool in logic and reasoning.
To check for the contrapositive of a statement, you simply switch the hypothesis and conclusion of the original statement and negate both. It is important to remember to switch the "if" and "then" parts of the statement and to use the word "not" to indicate negation.
Sure, an example of a contrapositive statement is "If a number is not divisible by 2, then it is not an even number." This is the contrapositive of the statement "If a number is an even number, then it is divisible by 2."
In science, we use logic and reasoning to make conclusions and predictions. Knowing how to form and check for contrapositive statements can help us to verify the validity of our hypotheses and theories. It can also help us to identify potential flaws in our reasoning and strengthen our arguments.