TIFR CSE 2012 | Part B | Question: 3

Question

For a person $p$, let $w(p)$, $A(p, y)$, $L(p)$ and $J(p)$ denote that $p$ is a woman, $p$ admires $y$, $p$ is a lawyer and $p$ is a judge respectively. Which of the following is the correct translation in first order logic of the sentence: "All woman who are lawyers admire some judge"?

• A. $\forall x: \left[\left(w\left(x\right)\Lambda L \left(x\right)\right)\Rightarrow \left(\exists y:\left(J \left(y\right)\Lambda w\left(y\right) \Lambda A\left(x, y\right)\right)\right)\right]$
• B. $\forall x: \left[\left(w\left(x\right)\Rightarrow L \left(x\right)\right)\Rightarrow \left(\exists y:\left(J \left(y\right) \Lambda A\left(x, y\right)\right)\right)\right]$
• C. $\forall x \forall y: \left[\left(w\left(x\right) \Lambda L\left(x\right)\right) \Rightarrow \left(J\left(y\right) \Lambda A\left(x, y\right)\right)\right]$
• D. $\exists y \forall x: \left[\left(w\left(x\right) \Lambda L\left(x\right)\right) \Rightarrow \left(J\left(y\right) \Lambda A\left(x, y\right)\right)\right]$
• E. $\forall x: \left[\left(w\left(x\right) \Lambda L\left(x\right)\right) \Rightarrow \left(\exists y: \left(J\left(y\right) \Lambda A\left(x, y\right)\right)\right)\right]$

Answer 1

Just translating to English:

1. Every women who is a lawyer admires some women judge.
2. If a person being women implies she is a lawyer then she admires some judge. OR If a person is not women or is a lawyer he/she admires some judge.
3. Every women who is a lawyer admires every judge.
4. There is some judge who is admired by every women lawyer.
5. Every women lawyer admire some judge. 

So, option (e) is the answer. 

Comments

nice explanation @Arjun sir

Answer 2

It will be (e).

Comments

i think e will be right still why d is wrong. both conclude the same thing. is this a multiple answer question ?

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d is wrong becz  it is

 ∃y∀x. It means All woman will admire same judge. But it should be other way.

∀X∃y. Then D will be true.

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∃y∀x and ∀x∃y are two differnt things.

∃y∀x thier exist some y for all x.

∀x∃y for all x thier exist a y.

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i got that one.

Answer 3

(e) is the correct translation.
∀x:[(w(x)ΛL(x))⇒(∃y:(J(y)ΛA(x,y)))]  :"For every person x, if x is woman AND lawyer then she admires some judge"  which is equivalent to say "Every women lawyer admire some judge" .

So, Ans is (e).

Answer 4

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