# Ways of writing a logical argument

• I
• Mr Davis 97
So they are equivalent.In summary, the map from ##G## to itself defined by ##\phi (g) = g^{-1}## being a homomorphism implies that ##G## is abelian, as shown by the fact that for any ##a,b \in G##, ##ab = b a##.
Mr Davis 97
Let ##G## be a group. Suppose that the map from ##G## to itself defined by ##\phi (g) = g^{-1}## is a homomorphism. Prove that ##G## is abelian.

So I came up with two ways of writing the solution and am wondering whether they are equivalent and which one is preferable:
1) Let ##a,b \in G##. Then ##\phi (ab) = \phi(a)(b) \implies (ab)^{-1} = a^{-1}b^{-1} \implies b^{-1}a^{-1} = a^{-1}b^{-1} \implies ba = ab##.

2) Let ##a,b \in G##. Then ##ab = (b^{-1}a^{-1})^{-1} = \phi(b^{-1}a^{-1}) = \phi(b^{-1}) \phi(a^{-1}) = b a##

It doesn't matter, they are both valid. I found the first one easier to read (you've forgotten one ##\phi##), but this is a matter of taste.

The basic idea is contained in both of them: inversion reverses the order whereas a homomorphism preserves the order. Both at the same time forces commutativity.

Mr Davis 97

## 1. What are the key elements of a logical argument?

The key elements of a logical argument include a clear and debatable claim, supporting evidence, logical reasoning, counterarguments, and a conclusion.

## 2. How do I structure a logical argument?

A logical argument is typically structured as follows: introduction (including the claim), body paragraphs (each with supporting evidence and reasoning), counterarguments, and conclusion (restating the claim and summarizing the main points).

## 3. What are some common flaws to avoid in a logical argument?

Some common flaws to avoid in a logical argument include using emotional appeals instead of logical reasoning, relying on biased or insufficient evidence, making generalizations or assumptions, and ignoring counterarguments.

## 4. How can I make my logical argument more persuasive?

To make your logical argument more persuasive, you can use a variety of techniques such as using strong and relevant evidence, anticipating and addressing counterarguments, using logical fallacies to strengthen your argument, and using persuasive language and tone.

## 5. Is there a specific format or style to follow when writing a logical argument?

While there is no specific format or style to follow, it is important to use a clear and organized structure, provide evidence to support your claims, use logical reasoning, and engage with potential counterarguments. Additionally, following a proper citation style and using clear and concise language can also enhance the effectiveness of your argument.

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