Question

Consider the following statement. For all positive real numbers r and s, vr + Str + VS. Some of the sentences in the following scrambled list can be used in a proof by contradiction for the statement. But this is a contradiction because r and s are positive. Simplifying the equation gives 0 = 2V75 | But this is a contradiction because r and s are negative. Squaring both sides of the equation gives r + s = r + 2yrs + s. Squaring both sides of the equation gives r + s = r + 2rs + s. By the zero product property, at least one of vror vs equals 0, which implies r or s equals 0. Construct a proof by contradiction of the statement by using the appropriate sentences from the list and putting them in the correct order. 1. Suppose not. That is, suppose there exists positive real numbers r and s such that r + s = ✓r + VS. 2. But this is a contradiction because r and s are positive. 3. ---Select--- 4. ---Select--- 5. --Select--- 6. Thus we have reached a contradiction and have proved the statement.

Answer 1

To construct a proof by contradiction for the given statement, assume the negation of the statement and show that it leads to a contradiction. Use the fact that when two positive numbers add, the answer has a positive sign.

Step-by-step explanation:

To construct a proof by contradiction for the given statement, we can follow the following steps:

1. Assume the negation of the statement, which is: There exist positive real numbers r and s such that vr + Str + VS.
2. Show that this assumption leads to a contradiction.
3. Since r and s are positive real numbers, we can use the fact that when two positive numbers add, the answer has a positive sign.
4. We can simplify the equation vr + Str + VS to obtain the contradiction 0 = 2V75.
5. By reaching this contradiction, we have proved the statement to be true since its negation led to an impossibility.