Concept explainers
(a)
To prove: every point inside the circle
(a)
Explanation of Solution
Given information:
Proof:
On the graph we can see that the claim is correct, let’s prove it formally.
Assume x and y are real numbers such that.
The proof is done.
(b)
To prove: if
(b)
Explanation of Solution
Given information:
Proof:
On the graph we can see that the claim is correct, let’s prove it formally.
Assume x and y are real numbers such that.
The proof is done.
(c)
To prove: every point inside the circle
(c)
Explanation of Solution
Given information:
Proof:
On the graph we can see that the claim is correct, let’s find a counterexample.
Let
It is in the first circle.
It is not in the second circle.
(d)
To prove: if
(d)
Explanation of Solution
Given information:
Proof:
On the graph we can see that the claim is correct, let’s find a counterexample.
Let
It is in the second circle.
But
(e)
To prove: if
(e)
Explanation of Solution
Given information:
Proof:
On the graph we can see that the claim is correct, let’s prove it formally.
Assume x and y are real numbers such that.
The proof is done.
(f)
To prove: there is a unique real number x such that for all real numbers y , if y < x then x + y is negative and x − y is positive.
(f)
Explanation of Solution
Given information: there is a unique real number x such that for all real numbers y, y < x.
Proof:
On the graph we can see that the claim is correct, let’s find a counterexample.
Let
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A Transition to Advanced Mathematics
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