13.3. Question 2

In this question, you are given two defined or two predicates which are defined for you. I(x) denotes that a stamp collector has stamp x in her collection and F(x, y) denotes that stamp x is issued by country y.

In this chunk, we introduce two predicates that represent specific conditions regarding a stamp collector's collection. 'I(x)' indicates whether the collector has a stamp 'x' in her collection. This predicate is key to expressing statements about the collector's possession of stamps. The second predicate, 'F(x, y)', shows the relationship between a stamp 'x' and the country 'y' that issued it. This introduction sets the stage for the statement we want to express about the collector having stamps from African countries.

Imagine you have a friend who collects coins instead of stamps. You could define 'C(x)' to indicate if your friend has coin 'x' and 'G(x, y)' to denote that coin 'x' is from country 'y'. Using these predicates, you can formulate statements about your friend's collection just like the stamp collector.

You have to express the statement that this collector has exactly one stamp issued by each African country. So, I am making a statement about a specific collector and I want to state that, for each African country, she has exactly one stamp issued by that country in her collection.

Here, we aim to articulate a specific criterion: that there should be exactly one stamp from each African country in the collector's possession. The phrase 'exactly one' is crucial because it implies two things: first, that at least one stamp exists for each country, and second, that there are no additional stamps from that same country. This dual requirement will be represented using logical quantifiers and predicates.

Consider a scenario where you want to ensure that your friend has exactly one unique coin from every state in the US. You would assert something similar: 'For every state, my friend has exactly one coin unique to that state.' This guarantees both that there's at least one coin and that there aren't duplicates from any state.

So if you recall from the lecture whenever we face this scenario where we want to represent a property p is true for exactly one element of the domain then there are two things which we have to represent.

When we need to express that a property is true for exactly one element in a given domain, we must ensure two conditions are met: first, that there is at least one such element, and second, that no other elements satisfy that property. This necessitates the use of both existential and universal quantifiers to construct a logical expression correctly representing this idea.

Think about a library where you want to highlight that each book can belong to only one specific genre. You need to establish that every genre has at least one book and there aren't multiple books classified under the same genre. It’s a matter of ensuring a clean, one-to-one correspondence.

For the moment forget about what is there in the remaining part of the expression forget it. Just focus on this part of the expression. But this is not what we want to represent because I cannot stop with this expression because this expression also means that there might be multiple x values for the same y, for the same country y.

In formal logic, simply stating that there exists a stamp for each African country is not sufficient if it does not rule out the possibility of multiple stamps from the same country. Therefore, our logical expression must also account for the absence of additional stamps, enforcing the condition of 'exactly one'. This leads us to refine our expression further to include a negation of any other stamps being present.

Imagine you’re organizing a school event where each class can only present one project. If you simply said that each class presents a project without ensuring that each class has only one, you could end up with multiple presentations from the same class, which would defeat the purpose.

That is why I put a negation here and if I put a negation that means there cannot be any other stamp x’ different from x, which is also there in the collection of the collector and x’ was issued by the African country y.

To fully express the condition of having exactly one stamp from each African country, we add a negation indicating that no other stamp from that same country can exist in the collection. This allows us to assert that not only is there at least one stamp present, but no duplicates exist for each country. This comprehensive approach ensures that the statement accurately reflects the collector's stamp collection.

Returning to our library example, the rule about books belonging to only one genre can be likened to stating that each genre cannot have more than one book being categorized under it. By emphasizing this, we ensure genre clarity and organization.

And that is why the conjunction of these two things represents the required statement. Of course, you can simplify this, apply the De Morgan’s law and take this negation inside convert everything, make everything in the form of an implication and so on that also you can do but even if you write this expression, that is correct.

By combining our two main conditions—with one ensuring at least one stamp exists and the other negating the presence of additional stamps— we form a conjunction that satisfies our requirements. This logical expression provides a clear and comprehensive statement about the collector's possession, ensuring it is both accurate and complete.

If you were to add the final organizational rule for your class presentations, you might say, 'Each class presents exactly one project with no repeats.' By clearly stating the conditions, you ensure that everyone understands the expectations and adheres to them.

--