IIT M video questions

Question

P(x,y,z),   xy=z, Universe is interger;

write in logic form

If xy=x for all y, then x =0.

Thank you

Answer 1

It's a simple "If----then" statement.

Hence, the formulation would be of type $A \rightarrow B$

Here, $A$ is   "xy=x for all y"...Or rephrasing, "For all y, xy = x"

Hence, $A$ = $\forall y (P(x,y,x))$

And $B$ is "$x = 0$"

So, "If xy=x for all y, then x =0" $\equiv$ $\forall y (P(x,y,x))$ $\rightarrow$ ($x = 0$)

You can check the above Propositional expression is Always True.

Comments

See this https://gateoverflow.in/74921/multiplicative-inverse#c221200

'=' operator is used here too

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'=' operator is used here too

 I have also used...see.. ∀y(P(x,y,x)) → (x=0)

Since, Universe is integer, You could use $x = 0$ if need be.

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Sir, what about  ${\forall y (P(x,y,x)) \to P(x,1,0)}$? Can it be a solution?

Answer 2

$\forall y\exists x\left (\left (P\left ( x ,y,z\right )=x\right )\rightarrow\left ( xy=x \right )\Lambda \left ( x=0 \right ) \right )$

Comments

What "=" stands for ??

And What will be the English interpretation of this expression?

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= is just for the interpretation of statement

May be a bracket need to clear the statement "If xy=x for all y, then x =0."

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P(x,y,z)=x

What does it mean? 

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ultimately we need to get x as result and we are operating on function P(x,y,z) i.e. P(x,y,z)=x

and for that conditions are (xy=x) and (x=0)

I think it is resolution method http://nptel.ac.in/courses/106106140/39