1. For all real numbers x, there is a real number y such that 2x+y=7 would this be true or false?

wstecznyg5 2022-07-15 Answered
Discrete math logic question
I have the following two questions.
1. For all real numbers x, there is a real number y such that 2 x + y = 7 would this be true or false?I think true because if you put
2 ( 7 ) + y = 14
2 ( 8 ) + y = 14 there will always be a specific y that will make it work is this logic correct.
1. There is a real numbers x that for all real number y, 2 x + y = 7 will be true.
would this be false because if you say x = 6
then you get
2 ( 6 ) + 2 = 14
only if y = 2 would it work but it would not work for every y.
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Answers (2)

Anaya Gregory
Answered 2022-07-16 Author has 14 answers
Step 1
Yes, you're on the right track. Both statements can be more easily analyzed by rearranging to get
x = 14 y 2
Step 2
This should allow you to fill in the gaps in your justification on (1) (i.e. not just stating a few examples in which it works).
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Tamara Bryan
Answered 2022-07-17 Author has 2 answers
Step 1
Yes, indeed, you are correct in your assessment of the truth or falsity of each statement.
In the first, we can see this as allowing y to depend on x. So for any given x, we can find some y, and in particular, we can simply choose y = 7 2 x which will guarantee the equalition holds.
Step 2
In the second case, y cannot depend on any given x. For the statement to be true, we need to consider the existence of a particular y such that for every x, regardless of what x may be, the equality holds. Since x can vary, but y can not vary accordingly, the statement is clearly false.
These two statements help demonstrate just how crucial the order of quantifiers and quantified variables can be: in the first, we have a true statement, and in the second, a false statement, and the only difference between them is the placement of y
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