Interpreting a statement in first order logic

  • #1
1,462
44

Homework Statement


Rewrite the following statements in symbolic form:

a) If ##a## and ##b## are real numbers with ##a \ne 0##, then ##ax+b=0## has a solution.
b) If ##a## and ##b## are real numbers with ##a \ne 0##, then ##ax+b=0## has a unique solution.

Homework Equations




The Attempt at a Solution



Attempts at solution:

Let ##P(x,a,b)## be the statement that ##ax+b=0## is true.
a) ##\forall a \in \mathbb{R} - \{0\} \forall b \in \mathbb{R} \exists x \in \mathbb{R} P(x,a,b)##
b) ##\forall a \in \mathbb{R} - \{0\} \forall b \in \mathbb{R} \exists x \in \mathbb{R} (P(x,a,b) \wedge \forall y (P(y,a,b) \implies y=x))##

Is that at all right? Is there an easier way? It all seems very cumbersome.
 

Answers and Replies

  • #2
15,107
12,784

Homework Statement


Rewrite the following statements in symbolic form:

a) If ##a## and ##b## are real numbers with ##a \ne 0##, then ##ax+b=0## has a solution.
b) If ##a## and ##b## are real numbers with ##a \ne 0##, then ##ax+b=0## has a unique solution.

Homework Equations




The Attempt at a Solution



Attempts at solution:

Let ##P(x,a,b)## be the statement that ##ax+b=0## is true.
a) ##\forall a \in \mathbb{R} - \{0\} \forall b \in \mathbb{R} \exists x \in \mathbb{R} P(x,a,b)##
b) ##\forall a \in \mathbb{R} - \{0\} \forall b \in \mathbb{R} \exists x \in \mathbb{R} (P(x,a,b) \wedge \forall y (P(y,a,b) \implies y=x))##

Is that at all right? Is there an easier way? It all seems very cumbersome.
Looks good to me, although I'm no logic expert, and I would add an ##\in \mathbb{R}## to the ##y##.
To make it less cumbersome a professor of mine used ##\exists !## to express uniqueness, but this isn't an official symbol. Some even used a symbol for "without loss of generality".
 
  • #3
member 587159
Looks good to me, although I'm no logic expert, and I would add an ##\in \mathbb{R}## to the ##y##.
To make it less cumbersome a professor of mine used ##\exists !## to express uniqueness, but this isn't an official symbol. Some even used a symbol for "without loss of generality".

Didn't know that this isn't an official symbol. I use it all the time.
 
  • #4
15,107
12,784
Didn't know that this isn't an official symbol. I use it all the time.
I don't know for certain, I only checked it in Wikipedia. But it makes sense, as it combines two statements in one. It is convenient, but not pure in a logical sense.
 

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