Question About Universal Quantifier & Logic

Question

Which are the correct equations? Due to my knowledge seems to be the 1st one is correct, but the others I can't figure out.

$$\begin{array} {lrcl} \text{(1).}~ & \forall x \in \mathbb R ~\forall y \in \mathbb R &:& y=2x+1 \iff x= \frac{y-1}{2} \\ \text{(2).}~ & \forall x \in \mathbb R ~\forall y \in \mathbb R &:& y^2-y+x^2-x=0 \iff (x=y \land x=0) \\ \text{(3).}~ & \forall x \in \mathbb R ~\forall y \in \mathbb R &:& (x=y \land x=0) \implies y^2-y+x^2-x=0 \\ \end{array}$$

Can anyone help me to solve this question? Thanks P.S.-(Sorry to say that above question is wrong due to my mistake.I re-edited the question.plz help!)

Answer 1

Instead of pointing out "correct equations", I will talk about "true formulas" among the three assuming that they are First Order formulas.

1. In this case we have, $$\boxed{\forall x\in\mathbb{R}\forall y\in\mathbb{R}\left(y=2x+1\iff x=\dfrac{(y-1)}{2}\right)}$$What does this formula actually mean? It means the following,

   For all $x,y\in\mathbb{R}$ if $y=2x-1$ then $x=\dfrac{y-1}{2}$ and vice versa.

   Do you think that it is right?

2. In this case we have, $$\boxed{\forall x\in\mathbb{R}\forall y\in\mathbb{R}\left(y^2−y+x^2−x=0⟺(x=y∧x=0)\right)}$$but observe that if you take $y=1=x$ then although $y^2−y+x^2−x=0$ still $x=y\land x=0$ is not true because $x\ne 0$. Hence this formula is false.

3. But observe that, $$\boxed{\forall x\in\mathbb{R}\forall y\in\mathbb{R}\left((x=y∧x=0)\implies y^2−y+x^2−x=0\right)}$$is true because the formula $(x=y\land x=0)\implies (y=0\land x=0)$. So if you substitute this value of $x$ and $y$ in the equation, you get $y^2-y+x^2-x=0$. Hence this formula is true to.

Answer 2

The first one is indeed correct.

But the two others don't make sense, because for instance

$$y^2-y+x^2-x$$

is not a valid proposition. It is not true or false, it is like saying

$$\pi.$$

Answer 3

The language of the first order logic consists of two main things: functional and predicate symbols (+auxiliaries like brackets).

From functional symbols $(+,-,...)$ terms are made: e.g. $x$ (variable) , $0$ (constant), $(x+y)$, $(x+y)-y$. These are interpreted as functions on elements of the domain of discourse.

Atomic formulas are then of the form $P(t_1,...t_n)$ for n-ary predicate symbol $P$ and terms $t_1,\dots t_n$. They expresed properties of elements. Examples: $x+y=z$, $x\leq y$,..(both $\leq, =$ are binary predicate symbols)

The well founded formulas are then obtained from atomic ones using propositional connectives $\vee, \wedge, \Rightarrow,\Leftrightarrow...$ and quantifiers.

An $equation$ is a first order formula of the form $\forall x_1..,x_n(t=s)$, where $s,t$ are terms with variables among $x_1,\dots x_n$.

Thus only the first expression of yours is a well founded formula: e.g. in the second one you cannot apply term $y^2-y+x^2-x$ to a connective $\Leftrightarrow$. Moreover, none of them is what is usually called an equation.