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I am a little confused about this sentence. How can I symbolize it?

Let P(x), L(x), R(x,y,z), and E(x,y) represent "x is a point," "x is a 
line," "z passes through x and y," and "x = y," respectively. Translate 
the following:

For every two points, there is one and only one line passing through 
both points.

$∀(x,y)((P(x) ∧ P(y) )→ (L(z) ∧ R(x,y,z)))$

so I want to say that for all x and y , if x and y are points then z is a line and it passes through x and y. is my formulae correct? I guess not because I am not using E(x, y) at all and I don't understand where to use it.

EDIT:

I think this is it:

$$ ∀(x,y)((P(x) ∧ P(y) )→(∃z(R(x,y,z) ∧ ∀k(R(x, y, k)) → E(z, k))) $$

Maktavish
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1 Answers1

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For every two points, there is one and only one line passing through both points.

$∀(x,y)~\big((P(x)∧P(y))→(∃z(R(x,y,z)∧∀k(R(x,y,k))→E(z,k)))\big)$

... is my formulae correct? I guess not because I am not using E(x, y) at all and I don't understand where to use it.

Yes, you also need to specify that the points are distinct (so there are two, rather than one). So that is, "For every $x,y$ that are points and distinct, then..."

Also, you must identify that the things they pass through need to be lines. Further, just as a suggestion to improve reader comprehension, rather than $z,k$ for lines, perhaps use alphabetically adjacent letters, say, $u,v$.

$$\small \forall x\forall y~\big((P(x)\land P(y)\land\lnot E(x,y))\to\exists u~(L(u)\land R(x,y,u)\land\forall v~((L(v)\land R(x,y,v))\to E(u,v)))\big)$$

You could also use the alternative definition of uniqueness, which is more compact.

$$\forall x~\forall y~\exists u~\forall v~\big((P(x)\land P(y)\land\lnot E(x,y))\to(E(u,v)\leftrightarrow (L(v)\land R(x,y,v))\big)$$

"For each distinct pair of points, there is something where anything is equal to it if and only if that is a line which passes through the points."

Graham Kemp
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