I'm new to proofs. Would someone please give me an opinion on my proof?

In summary, the conversation discussed the use of twelve basic properties of numbers in a proof. These properties include the associative law for addition, additive identity, additive inverse, commutative law for addition, associative law for multiplication, multiplicative identity, multiplicative inverse, commutative law for multiplication, distributive law, trichotomy law, closure under addition, and closure under multiplication. The conversation also included an attempt at a proof using these properties to show that if 0 < a < b, then a < √(ab) < (a+b)/2 < b. Suggestions were given to use the properties more explicitly and to simplify certain steps in the proof.
  • #1
nietzsche
186
0
My prof gave us twelve basic properties of numbers, and I think I'm supposed to use those in my proof, but I'm not sure how to incorporate them.

The properties are:
P1 Associative law for addition
P2 Additive identity
P3 Additive inverse
P4 Commutative law for addition
P5 Associative law for multiplication
P6 Multiplicative identity
P7 Multiplicative inverse
P8 Commutative law for multiplication
P9 Distributive law
P10 Trichotomy law
P11 Closure under addition
P12 Closure under multiplication

Homework Statement



Prove that if 0 < a < b, then

[tex]a < \sqrt{ab} < \frac{a + b}{2} < b[/tex]

Homework Equations



N/A

The Attempt at a Solution



Part I:

[tex]
\begin{align*}
a&<b\\
a^2&<ab\\
\sqrt{a^2}&<\sqrt{ab}\\
a&<\sqrt{ab}
\end{align*}
[/tex]

Part II:

[tex]
Suppose:
\begin{align*}
\sqrt{ab}&\geq\frac{a+b}{2}\\
ab&\geq\left(\frac{a+b}{2}\right)^2\\
ab&\geq\frac{a^2+2ab+b^2}{4}\\
4ab&\geq a^2+2ab+b^2\\
0&\geq a^2-2ab+b^2\\
0&\geq (a-b)^2\\
\end{align*}
[/tex]

[tex]
but:
\begin{align*}
0&<(a-b)^2\\
\therefore \sqrt{ab}&<\frac{a+b}{2}
\end{align*}
[/tex]

Part III:

[tex]
\begin{align*}
a&<b\\
a+b &< b+b\\
a+b &< 2b\\
\frac{a+b}{2} &< b
\end{align*}
[/tex]

So I'm able to prove them, but I don't know if I used the properties correctly (if at all). Any opinions or suggestions?
 
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  • #2
nietzsche said:
My prof gave us twelve basic properties of numbers, and I think I'm supposed to use those in my proof, but I'm not sure how to incorporate them.

The properties are:
P1 Associative law for addition
P2 Additive identity
P3 Additive inverse
P4 Commutative law for addition
P5 Associative law for multiplication
P6 Multiplicative identity
P7 Multiplicative inverse
P8 Commutative law for multiplication
P9 Distributive law
P10 Trichotomy law
P11 Closure under addition
P12 Closure under multiplication

Homework Statement



Prove that if 0 < a < b, then

[tex]a < \sqrt{ab} < \frac{a + b}{2} < b[/tex]

Homework Equations



N/A

The Attempt at a Solution



Part I:

[tex]
\begin{align*}
a&<b\\
a^2&<ab\\
\sqrt{a^2}&<\sqrt{ab}\\
a&<\sqrt{ab}
\end{align*}
[/tex]
By "using the properties" I think your prof means for you to indicate which property allows you to do each step. However, in some of your steps you are using operations that aren't listed amongst the properties you show. For example, in your 2nd inequality, when you multiply both members of an inequality by a positive number, the direction of the inequality stays the same.
nietzsche said:
Part II:

[tex]
Suppose:
\begin{align*}
\sqrt{ab}&\geq\frac{a+b}{2}\\
ab&\geq\left(\frac{a+b}{2}\right)^2\\
ab&\geq\frac{a^2+2ab+b^2}{4}\\
4ab&\geq a^2+2ab+b^2\\
0&\geq a^2-2ab+b^2\\
0&\geq (a-b)^2\\
\end{align*}
[/tex]

[tex]
but:
\begin{align*}
0&<(a-b)^2\\
\therefore \sqrt{ab}&<\frac{a+b}{2}
\end{align*}
[/tex]
For the one above, instead of doing a proof by contradiction, as you have done, it would be simpler to start with (a - b)2 >= 0 (the square of any real number is always nonnegative). Then expand the left side and you should be able to get to the conclusion you need.
nietzsche said:
Part III:

[tex]
\begin{align*}
a&<b\\
a+b &< b+b\\
a+b &< 2b\\
\frac{a+b}{2} &< b
\end{align*}
[/tex]

So I'm able to prove them, but I don't know if I used the properties correctly (if at all). Any opinions or suggestions?
In the one above, b + b = b(1 + 1) = b*2 = 2b. The properties used are the distributive property and the commutative property of multiplication.
 
  • #3


I would suggest that you review each property and try to incorporate them into your proof. For example, you could use the associative law for multiplication (P5) to rearrange the terms in Part II of your proof. You could also use the additive identity (P2) and the distributive law (P9) to simplify your equations in Part II. Additionally, you could use the closure under multiplication (P12) to show that ab is a positive number, which is necessary for taking its square root. By incorporating these properties, you can strengthen your proof and demonstrate a deeper understanding of the concepts. I would also recommend discussing your proof with your professor or classmates to get feedback and improve upon it. Keep practicing and you will become more comfortable with using these properties in your proofs.
 

1. How can I improve my proof?

One way to improve your proof is to carefully review and analyze each step to ensure that it logically follows from the previous step. You can also ask someone else to review your proof and provide feedback.

2. Are there any common mistakes to avoid in proofs?

Some common mistakes to avoid in proofs include assuming what you are trying to prove, using circular reasoning, and not providing enough justification for each step.

3. How do I know if my proof is correct?

A correct proof is one that follows the rules of logic and provides a logical and convincing argument for the statement being proven. You can also double check your proof by using a proof checker or asking someone else to review it.

4. How can I make my proof more concise?

To make your proof more concise, try to eliminate any unnecessary steps or explanations. You can also use symbols or shorthand notation to represent longer words or phrases.

5. What should I do if I get stuck on a proof?

If you get stuck on a proof, try breaking it down into smaller parts and tackling each part individually. You can also consult textbooks or online resources for examples or guidance. Don't be afraid to ask for help from a teacher or classmate as well.

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