Suppose P is the predicate defined by P(x, y) = x is friends with y, where x and y are people. (No one is considered to be friends with themselves.) Translate the formal expression ∀x∃y∃z(y ≠ z ∧ P(x, y) ∧ P(x, z)) into English.
Answer (2)
The given formal expression ∀ x ∃ y ∃ z ( y = z ∧ P ( x , y ) ∧ P ( x , z )) can be translated into English as follows:
"For every person x , there exist two different people y and z such that x is friends with both y and z ."
Let's break this down step-by-step to better understand the expression:
∀ x : This part means "for every person x ." We are considering every individual x in the context we're discussing—imagine a group of people, and x represents each person one at a time.
∃ y ∃ z : This translates to "there exist people y and z ." It indicates that we are looking to find specific people y and z for each person x .
y = z : This tells us that y and z must be different people. x cannot be friends with the same person twice, so y and z need to be distinct individuals.
P ( x , y ) ∧ P ( x , z ) : The predicate P ( x , y ) means " x is friends with y ," and P ( x , z ) means " x is friends with z ." The ∧ symbol stands for 'and,' meaning both statements must be true simultaneously.
In summary, the expression asserts that no matter whom you pick in a certain group, that person ( x ) has at least two different friends ( y and z ). This statement is often analyzed in the context of social networks or graph theory to understand relationships and social connections in mathematical terms.
The formal expression asserts that for every person, there are at least two distinct friends they have. This indicates that in a given social group, each individual maintains relationships with multiple others. It reflects on the connectedness of people in social networks.
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