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I have started learning lambda calculus from the book by Hindley and Seldin  .

It brought up the concept with the function 'x-y'First$x-y$, first as a function of "x"$x$ and then as a function of "y"$y$.In In a way  , it emphasized the difference between treating the expression as a function of "x:$x$ and function of "y"$y$,giving giving separate names to the functions  .

$f(x,y) = x -y $ and $g(y,x) = x -y $$$f(x,y) = x-y \qquad\text{and}\qquad g(y,x) = x -y$$ and, in $\lambda$ notion

$ h = \lambda x.x-y $ and $g = \lambda y x. x-y $, $$h = \lambda xy.x-y \qquad\text{and}\qquad g = \lambda yx. x-y\,.$$

If I am not wrong then it can be inferred that :

$$f =\lambda x .(x-y) $$$f =\lambda x .(x-y)$ means that while computing the value of the expression the expression will vary only with "x" $x$ and similarly when "lambda "$\lambda$ is placed with $ x $$x$ the other variable $y$" will be kept as a constant  .  

But then it said that this can be denoted as  ,with with $h$ being the common name of the function

$h = \lambda xy.(x-y) $ and $h = \lambda yx.(x-y) $ $$h = \lambda xy.(x-y)\qquad\text{and}\qquad h = \lambda yx.(x-y)\,.$$
Is it so that in the above expression the variable which is adjacent to $\lambda$ receives the value and the other one is constant  ? The above function is called a two-place function ,whatfunction; what does itthat mean  ? Then

Then he introduces the following function in $\lambda$ notionnotation calling it a one point-point function  : $h^\star = \lambda x .(\lambda y . x−y)$$$h^\star = \lambda x .(\lambda y . x−y)$$ and mentions the following

Forsays that, for each number a$a$, we have $h^\star(a)=λy . a−y$ $h^\star(a)=\lambda y . a−y$. Here $a$ is being provided as an argument but $\lambda$ is placed near $y$  . Why sois this? And

And then it deduces the followingthat :

$(h^\star(a))(b)=(\lambda y . a−y)(b) = a−b = h(a,b)$ and says $h^\star$ can be viewed as 'representing '"representing" $h$  . Is "representing" a technical term  ?How How was $(\lambda y . a−y)(b) = a−b$ concluded .

P.S : I am afraid this question may be classified as unclear or broad but I hope that this can be unified as a problem of getting though the $\lambda$ notion or missing something and the few questions asked at different parts if this post will have very closely related answer.?

I have started learning lambda calculus from the book by Hindley and Seldin  .

It brought up the concept with the function 'x-y'First as a function of "x" and then as a function of "y".In a way  , it emphasized the difference between treating the expression as a function of "x: and function of "y",giving separate names to the functions  .

$f(x,y) = x -y $ and $g(y,x) = x -y $ and in $\lambda$ notion

$ h = \lambda x.x-y $ and $g = \lambda y x. x-y $

If I am not wrong then it can be inferred that :

$$f =\lambda x .(x-y) $$ means that while computing the value of the expression the expression will vary only with "x" and similarly when "lambda " is placed with $ x $ the other variable $y$" will be kept as a constant  .  

But then it said that this can be denoted as  ,with $h$ being the common name of the function

$h = \lambda xy.(x-y) $ and $h = \lambda yx.(x-y) $
Is it so that in the above expression the variable which is adjacent to $\lambda$ receives the value and the other one is constant  ? The above function is called a two-place function ,what does it mean  ? Then he introduces the following function in $\lambda$ notion calling it a one point function  : $h^\star = \lambda x .(\lambda y . x−y)$ and mentions the following

For each number a, we have $h^\star(a)=λy . a−y$ . Here $a$ is being provided as an argument but $\lambda$ is placed near $y$  . Why so? And then it deduces the following :

$(h^\star(a))(b)=(\lambda y . a−y)(b) = a−b = h(a,b)$ and says $h^\star$ can be viewed as 'representing ' $h$  . Is "representing" a technical term  ?How was $(\lambda y . a−y)(b) = a−b$ concluded .

P.S : I am afraid this question may be classified as unclear or broad but I hope that this can be unified as a problem of getting though the $\lambda$ notion or missing something and the few questions asked at different parts if this post will have very closely related answer.

I have started learning lambda calculus from the book by Hindley and Seldin.

It brought up the concept with the function $x-y$, first as a function of $x$ and then as a function of $y$. In a way, it emphasized the difference between treating the expression as a function of $x$ and function of $y$, giving separate names to the functions.

$$f(x,y) = x-y \qquad\text{and}\qquad g(y,x) = x -y$$ and, in $\lambda$ notion, $$h = \lambda xy.x-y \qquad\text{and}\qquad g = \lambda yx. x-y\,.$$

If I am not wrong then it can be inferred that $f =\lambda x .(x-y)$ means that while computing the value of the expression the expression will vary only with $x$ and similarly when $\lambda$ is placed with $x$ the other variable $y$ will be kept as a constant.

But then it said that this can be denoted as, with $h$ being the common name of the function $$h = \lambda xy.(x-y)\qquad\text{and}\qquad h = \lambda yx.(x-y)\,.$$
Is it so that in the above expression the variable which is adjacent to $\lambda$ receives the value and the other one is constant? The above function is called a two-place function; what does that mean?

Then he introduces the following function in $\lambda$ notation calling it a one-point function: $$h^\star = \lambda x .(\lambda y . x−y)$$ and says that, for each number $a$, we have $h^\star(a)=\lambda y . a−y$. Here $a$ is being provided as an argument but $\lambda$ is placed near $y$. Why is this?

And then it deduces that $(h^\star(a))(b)=(\lambda y . a−y)(b) = a−b = h(a,b)$ and says $h^\star$ can be viewed as "representing" $h$. Is "representing" a technical term? How was $(\lambda y . a−y)(b) = a−b$ concluded?

2 Fixed h "exponent" in LaTeX
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I have started learning lambda calculus from the book by Hindley and Seldin .

It brought up the concept with the function 'x-y'First as a function of "x" and then as a function of "y".In a way , it emphasized the difference between treating the expression as a function of "x: and function of "y",giving separate names to the functions .

$f(x,y) = x -y $ and $g(y,x) = x -y $ and in $\lambda$ notion

$ h = \lambda x.x-y $ and $g = \lambda yx. x-y $$g = \lambda y x. x-y $

If I am not wrong then it can be inferred that :

$$f =\lambda x .(x-y) $$ means that while computing the value of the expression the expression will vary only with "x" and similarly when "lambda " is placed with $ x $ the other variable $y$" will be kept as a constant .

But then it said that this can be denoted as ,with $h$ being the common name of the function

$h = \lambda xy.(x-y) $ and $h = \lambda yx.(x-y) $
Is it so that in the above expression the variable which is adjacent to $\lambda$ receives the value and the other one is constant ? The above function is called a two-place function ,what does it mean ? Then he introduces the following function in $\lambda$ notion calling it a one point function : $h* = λx .(λy . x−y)$$h^\star = \lambda x .(\lambda y . x−y)$ and mentions the following

For each number a, we have $h*(a)=λy . a−y$$h^\star(a)=λy . a−y$ . Here $a$ is being provided as an argument but $\lambda$ is placed near $y$ . Why so  ? And then it deduces the following : $(h*(a))(b)=(λy . a−y)(b) = a−b = h(a,b)$

$(h^\star(a))(b)=(\lambda y . a−y)(b) = a−b = h(a,b)$ and says $h*$$h^\star$ can be viewed as 'representing ' $h$ . Is "representing" a technical term ?How was $(λy . a−y)(b) = a−b$$(\lambda y . a−y)(b) = a−b$ concluded .

P.S : I am afraid this question may be classified as unclear or broad but I hope that this can be unified as a problem of getting though the $\lambda$ notion or missing something and the few questions asked at different parts if this post will have very closely related answer.

I have started learning lambda calculus from the book by Hindley and Seldin .

It brought up the concept with the function 'x-y'First as a function of "x" and then as a function of "y".In a way , it emphasized the difference between treating the expression as a function of "x: and function of "y",giving separate names to the functions .

$f(x,y) = x -y $ and $g(y,x) = x -y $ and in $\lambda$ notion

$ h = \lambda x.x-y $ and $g = \lambda yx. x-y $

If I am not wrong then it can be inferred that :

$$f =\lambda x .(x-y) $$ means that while computing the value of the expression the expression will vary only with "x" and similarly when "lambda " is placed with $ x $ the other variable $y$" will be kept as a constant .

But then it said that this can be denoted as ,with $h$ being the common name of the function

$h = \lambda xy.(x-y) $ and $h = \lambda yx.(x-y) $
Is it so that in the above expression the variable which is adjacent to $\lambda$ receives the value and the other one is constant ? The above function is called a two-place function ,what does it mean ? Then he introduces the following function in $\lambda$ notion calling it a one point function : $h* = λx .(λy . x−y)$ and mentions the following

For each number a, we have $h*(a)=λy . a−y$ . Here $a$ is being provided as an argument but $\lambda$ is placed near $y$ . Why so  ? And then it deduces the following : $(h*(a))(b)=(λy . a−y)(b) = a−b = h(a,b)$ and says $h*$ can be viewed as 'representing ' $h$ . Is "representing" a technical term ?How was $(λy . a−y)(b) = a−b$ concluded .

P.S : I am afraid this question may be classified as unclear or broad but I hope that this can be unified as a problem of getting though the $\lambda$ notion or missing something and the few questions asked at different parts if this post will have very closely related answer.

I have started learning lambda calculus from the book by Hindley and Seldin .

It brought up the concept with the function 'x-y'First as a function of "x" and then as a function of "y".In a way , it emphasized the difference between treating the expression as a function of "x: and function of "y",giving separate names to the functions .

$f(x,y) = x -y $ and $g(y,x) = x -y $ and in $\lambda$ notion

$ h = \lambda x.x-y $ and $g = \lambda y x. x-y $

If I am not wrong then it can be inferred that :

$$f =\lambda x .(x-y) $$ means that while computing the value of the expression the expression will vary only with "x" and similarly when "lambda " is placed with $ x $ the other variable $y$" will be kept as a constant .

But then it said that this can be denoted as ,with $h$ being the common name of the function

$h = \lambda xy.(x-y) $ and $h = \lambda yx.(x-y) $
Is it so that in the above expression the variable which is adjacent to $\lambda$ receives the value and the other one is constant ? The above function is called a two-place function ,what does it mean ? Then he introduces the following function in $\lambda$ notion calling it a one point function : $h^\star = \lambda x .(\lambda y . x−y)$ and mentions the following

For each number a, we have $h^\star(a)=λy . a−y$ . Here $a$ is being provided as an argument but $\lambda$ is placed near $y$ . Why so? And then it deduces the following :

$(h^\star(a))(b)=(\lambda y . a−y)(b) = a−b = h(a,b)$ and says $h^\star$ can be viewed as 'representing ' $h$ . Is "representing" a technical term ?How was $(\lambda y . a−y)(b) = a−b$ concluded .

P.S : I am afraid this question may be classified as unclear or broad but I hope that this can be unified as a problem of getting though the $\lambda$ notion or missing something and the few questions asked at different parts if this post will have very closely related answer.

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Basic notation in lambda calculus

I have started learning lambda calculus from the book by Hindley and Seldin .

It brought up the concept with the function 'x-y'First as a function of "x" and then as a function of "y".In a way , it emphasized the difference between treating the expression as a function of "x: and function of "y",giving separate names to the functions .

$f(x,y) = x -y $ and $g(y,x) = x -y $ and in $\lambda$ notion

$ h = \lambda x.x-y $ and $g = \lambda yx. x-y $

If I am not wrong then it can be inferred that :

$$f =\lambda x .(x-y) $$ means that while computing the value of the expression the expression will vary only with "x" and similarly when "lambda " is placed with $ x $ the other variable $y$" will be kept as a constant .

But then it said that this can be denoted as ,with $h$ being the common name of the function

$h = \lambda xy.(x-y) $ and $h = \lambda yx.(x-y) $
Is it so that in the above expression the variable which is adjacent to $\lambda$ receives the value and the other one is constant ? The above function is called a two-place function ,what does it mean ? Then he introduces the following function in $\lambda$ notion calling it a one point function : $h* = λx .(λy . x−y)$ and mentions the following

For each number a, we have $h*(a)=λy . a−y$ . Here $a$ is being provided as an argument but $\lambda$ is placed near $y$ . Why so ? And then it deduces the following : $(h*(a))(b)=(λy . a−y)(b) = a−b = h(a,b)$ and says $h*$ can be viewed as 'representing ' $h$ . Is "representing" a technical term ?How was $(λy . a−y)(b) = a−b$ concluded .

P.S : I am afraid this question may be classified as unclear or broad but I hope that this can be unified as a problem of getting though the $\lambda$ notion or missing something and the few questions asked at different parts if this post will have very closely related answer.