Step-by-Step Analysis

To find the expression that means "N is definitely daughter of K", we must first decode the symbols and then analyze each option to see if it satisfies the two necessary conditions: (1) K is a parent of N, and (2) N is female.

1. Symbol Definitions

• P ψ Q means P is the mother of Q. (P is female)
• P ∈ Q means P is the sister of Q. (P is female)
• P \ Q means P is the father of Q. (P is male)
• P # Q means P is the brother of Q. (P is male)

2. Evaluating the Options

We need an expression that makes K a parent of N and also defines N's gender as female. N's gender can only be determined if N appears as the first element in a relationship using the symbols ψ (mother) or ∈ (sister).

A. K \ L # M # N

• This translates to: K is the father of L, L is the brother of M, and M is the brother of N.
• This makes K the father of N, satisfying the first condition.
• However, N's gender is not defined. N could be a son or a daughter. Thus, it is not "definitely daughter".

B. M ψ K \ N ∈ L

• This translates to: M is the mother of K, K is the father of N, and N is the sister of L.
• K \ N means K is the father of N. This satisfies the parent condition.
• N ∈ L means N is the sister of L. This establishes that N is female.
• Since both conditions are met, this expression means N is definitely the daughter of K.

C. K ψ M # L ∈ N

• This translates to: K is the mother of M, M is the brother of L, and L is the sister of N.
• This makes K the mother of N, satisfying the parent condition.
• However, L ∈ N only tells us L's gender, not N's. N's gender remains unknown.

D. L ψ K \ N # M

• This translates to: L is the mother of K, K is the father of N, and N is the brother of M.
• N # M establishes that N is male, which contradicts the requirement that N be a daughter.

Conclusion

The only expression that definitively establishes N as female and K as her parent is M ψ K \ N ∈ L.