Proof in predicate calculus

In summary, we have proven that for all a and for all b, if a>0 and b>0, then a<=b if and only if a^2<=b^2. This can be shown using the axioms and theorems of predicate calculus, by substituting suitable terms for the variables and using logical equivalences.
  • #1
poutsos.A
102
1
[tex]\forall a\forall b[/tex][( a>0 & b>0)------> (a[tex]\leq b[/tex] <------>[tex]a^{2}[/tex][tex]\leq b^{2}[/tex])].

or in words: for all a and for all b , if a>0 and b>0 then .a[tex]\leq b[/tex] iff [tex]a^{2}[/tex][tex]\leq b^{2}[/tex]

is there a possibility for a proof within the predicate calculus??
 
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  • #2
Surely that's not what you meant? Surely you mean [itex]a^2\le b^2[/itex].
 
  • #3
poutsos.A said:
[tex]\forall a\forall b[/tex][( a>0 & b>0)------> (a[tex]\leq b[/tex] <------>[tex]a^{2}[/tex][tex]\leq b^{2}[/tex])].

or in words: for all a and for all b , if a>0 and b>0 then .a[tex]\leq b[/tex] iff [tex]a^{2}[/tex][tex]\leq b^{2}[/tex]

is there a possibility for a proof within the predicate calculus??

yes thank you it is [tex]a^{2}[/tex][tex]\leq b^{2}[/tex],instead of [tex]a^{2}[/tex][tex]\leq a^{2}[/tex]
 
  • #4
I'm not at all clear on what your question is.

How do you define "<" in the "predicate calculus"?
 
  • #5
< is a two place predicate symbol x<y or y<x .

The axioms and theorems concerning < are exactly those of the real Nos,i.e trichotomy law ,transitivity e.t.c.

Now writing a proof within predicate calculus means writing a proof where one uses quantified formulas in which the introduction and elimination of quantifiers is done according to the axioms and theorems of predicate calculus.

Note predicate calculus includes propositional logic as well
 
  • #6
[tex]\forall x\alpha\rightarrow\alpha^{x}_{t}[/tex], where t is substituable for x in [tex]\alpha[/tex].[tex]\alpha^{x}_{t}[/tex] is the expression obtained from the fomula [tex]\alpha[/tex] by replacing the variable x, whenever it occurs free in [tex]\alpha[/tex], by the term t (see "substitution section" of First-order logic (wiki).

[tex]\forall a\forall b\phi[/tex], where [tex]\phi[/tex] is the wff [( a>0 & b>0)------> (a<= b <-> a^2 <= b^2)], a and b occurs free in [tex]\phi[/tex]. Thus we can replace "a" by a term( variable) "x" and "b" by a term(variable) "y", respectively.

Now, it reduces to prove the below statement.
( x>0 & y>0)------> (x <=y <-> x^2 <= y^2).

To prove ->
Let y = x + k (k>=0) . Square y and compare x^2.
To prove <-
Use a contrapositive method.
 
  • #7
I don't see what there is to prove here, it's like p&~p contradiction, and pv~p a tautology. (-:

anyway, a<=b means there exists a real c>=0 s.t b=a+c, after squaring you get:
b^2=a^2+c^2+2ac, a is positive and so is c, denote d=c^2+2ac>0.
for the other way around you just reverse what youv'e done so far cleverly.
b^2=a^2+d, b=+-sqrt(a^2+d) we know that b>0 so we choose the plus sign, b=sqrt(a^2+d), now we need to show there exists e>0 such that b=a+e choose a+e=sqrt(a^2+d)
then raise the square, and I let you fill the minor details... (-:
 
  • #8
Correction, e=sqrt(a^2+d)-a

QED.
cheers...
 
  • #9
definitely the above proofs are not proofs within predicate calculus,perhaps an example from my notes will give you an idea.

thanks
 
  • #10
So how did you prove it eventually?
 

1. What is predicate calculus?

Predicate calculus is a formal mathematical system used to represent and reason about propositions involving quantifiers, variables, and predicates. It is a fundamental branch of mathematical logic and is widely used in various fields such as computer science, linguistics, and philosophy.

2. What is the purpose of proof in predicate calculus?

The purpose of proof in predicate calculus is to provide a rigorous and systematic way of demonstrating the validity of logical arguments. It allows us to determine whether a given proposition can be derived from a set of axioms and rules of inference, thus ensuring the correctness of our reasoning.

3. What are the basic components of a proof in predicate calculus?

A proof in predicate calculus typically consists of a set of assumptions, a sequence of logical steps, and a conclusion. The assumptions are the starting point of the proof, while the logical steps are used to derive new propositions from the given assumptions. The conclusion is the final statement that is proven to be true based on the initial assumptions and logical steps.

4. How is a proof in predicate calculus structured?

A proof in predicate calculus follows a formal structure, often referred to as a proof tree. The proof tree starts with the initial assumptions at the top and branches out into smaller subproofs, each with its own set of assumptions and logical steps. The subproofs eventually converge back to the main proof, leading to the final conclusion.

5. What are some common proof techniques used in predicate calculus?

Some common proof techniques used in predicate calculus include direct proof, proof by contradiction, proof by induction, and proof by contraposition. Each of these techniques involves using logical steps and rules of inference to demonstrate the validity of a proposition. They may also involve breaking down a complex proposition into simpler subpropositions to make the proof more manageable.

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