Question

(a) Write the contrapositive of the following statement: For all positive real numbers a and b, if √ab ≠ a+b/2 then a ≠ b

(b) Is this statement true or false? Prove the statement if it is true or provide a counterexample if it is false.

Answer 1

The contrapositive of the statement is provided along with a counterexample to show that the statement is false.

Step-by-step explanation:

The contrapositive of the statement "For all positive real numbers a and b, if √ab ≠ a+b/2 then a ≠ b" is "For all positive real numbers a and b, if a = b then √ab = a+b/2."

To prove if this statement is true or false, we can provide a counterexample. Let's use a = 4 and b = 4 as an example. Plugging these values into the original statement, we get √(4*4) ≠ 4+4/2, which simplifies to 4 ≠ 6. Since this is true, the original statement is false.