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I am learning lambda calculus and I was given the following assignment:

Working with these definitions of TRUE and FALSE:

λx y . x ≡ T

λx y . y ≡ F

I am asked to form the AND operator, that even though functions similarly, is not equivalent to the following definition of it:

∧ ≡ λx y .x y F

But everywhere I've checked this seems to be the only way to go at it. Can anyone provide any guidance as to how I should start thinking about it?

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    What does"equivalent" mean here exactly? The first thing I'd want to look at would be, how similar my solution be and still be considered"inequivalent"? – 2017-01-08
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    Not syntactically the same I suppose. – 2017-01-08
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    In that case, my hint is: the AND operation is commutative. – 2017-01-08
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    Does this mean that: ∧ ≡ λx y .y x F would be a sufficient solution? Looking at it it seems too simple to be right. Weird. – 2017-01-08
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    [This](https://en.wikipedia.org/wiki/Church_encoding#Church_Booleans) Wikipedia article contains alternative encoding. – 2017-01-08
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    I don't know if $ λx y .y x F$ would be a sufficient solution, because it turns on the meaning of “inequivalent”. If it means “not syntactically the same”, as you suggested, then it's fine. Other things you could consider are: $\eta$-expansion of the formula you have; or rewriting $x\land y$ as $\text{if } x \text{ then } x \text{ else } y$. – 2017-01-08
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    I meant "if y then x else y". – 2017-01-08
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    I just learned what exactly they meant with "inequivalent". It basically means, that if we were to plug in each function something other than boolean values, they would have different outcomes. – 2017-01-08

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