11.3. Without Loss of Generality (w.l.o.g.)

There is also something called without loss of generality, which we very often encounter while writing the proof. In short form they call it as w.l.o.g.

The term 'without loss of generality' (often abbreviated as w.l.o.g.) is used in mathematical proofs to simplify the argument being made. When we say 'without loss of generality,' we are indicating that the proof can be made easier or more manageable by examining a specific case, while knowing that the results will extend to other cases as well.

Imagine a scenario where you are trying to prove that a particular rule in a sports game applies whether a team is playing at home or away. You decide to analyze just one case — say, the home game — and state that since the rules are the same, this analysis will apply to away games too. By focusing on one situation without losing the generality of the argument, you simplify your proof.

So imagine I want to prove that if x and y are integers and both x, y and x + y are even, then x and y are both even.

Here, we want to demonstrate a statement about integers x and y. To use w.l.o.g. effectively, we can consider only one case instead of analyzing all possible cases separately. Given that both x and y are integers, we can assume without loss of generality that x is the odd number. If this assumption leads to a contradiction or proves our statement, we can conclude the validity of the overall statement without examining the alternative configurations.

Think about cooking. If you want to create a recipe and you know one ingredient works well, you might say, 'Let's assume I'm using chicken.' Even if you could use other meats, focusing on chicken makes the explanation clearer without affecting the overall point of the dish. Therefore, you establish your argument based on this chosen ingredient while knowing the outcome would be similar for others.

I can club both cases 1 and 2 into one new case 1, where I say that exactly 1 of x and y is odd which one I do not know.

By combining both cases into one case using w.l.o.g., we reduce complexity and avoid redundancy in our proof. Instead of treating cases where x could be odd and y even, or x even and y odd, we simplify to just asserting: 'Exactly one of them is odd.' This assumption does not lose generality because all earlier cases fall under this broader condition.

Imagine you are organizing a game night with friends. Instead of detailing the rules separately for every two-player team configuration, you could just explain, 'As long as one player is playing a strategy while the other is defensive, the dynamics will remain the same.' This simplification captures all configurations without losing important insights.

This argument automatically subsumes the case when y is odd as well and now my second case will be when both x and y are odd.

The implication here is that by using w.l.o.g., we automatically cover more cases under one umbrella term. Since we demonstrated the conclusions hold for our aggregated case—and given symmetry in the properties of x and y—it suffices for us to conclude about both variables. The efficiency gained in using w.l.o.g. is significant in simplifying proofs.

Think of a class where you’re assessing whether students prefer one type of study method over another. If you know one group of students favors a method, you might argue 'without loss of generality' that the opposing group will exhibit the same preferences. So, reviewing just one group's dynamics suffices to draw conclusions about preferences because the conditions apply to both.

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