Complete this formal proof of Ex(P(x)v~P(x)) from the empty set. NOTE: similar to the rule above...

Complete this formal proof of Ex(P(x)v~P(x)) from the empty set. NOTE: similar to the rule above when instantiating quantifiers, if you need a random name, always start at the beginning of the alphabet. That is, use a first; only use b if necessary; etc.

Expert Solution

Some proofs of definitions by contradiction and contrapositive but not direct proofs (the existence of infinite primes for example),
I think most of them because the direct proof extends away from a first mathematics course or the proofs by contradiction/contrapositive are more didactic.
The one that most bothers me in particular is the demonstration that the empty set is a subset of every set, and it is unique.
I understand the uniqueness and understand the proof by contradiction:

Suppose ∅⊊A where A is a set. So it exists an element x∈∅ such that x∉A wich is absurd because ∅ does not have any elements by definition.

but I would like to know if there exists a direct proof of this and if indeed extends from a first course.

ekkarill92 answered 7 months ago

Give a formal proof for the following tautology by using the CP rule. A v B→(¬...

Give a formal proof for the following tautology by using the CP rule. A v B→(¬ B →(A ^ ¬ B))

Let X be a non-empty set and P(X) its power set. Then (P(x), symetric difference, intersection)...

Let X be a non-empty set and P(X) its power set. Then (P(x), symetric difference, intersection) is a ring. Find a non-trivial ideal of P(X).

if A union B = S, A intersect B = empty set, P(A) = x, P(B)...

if A union B = S, A intersect B = empty set, P(A) = x, P(B) = y, and 3x-y = 1/2, find x and y.

A linear transformation from R3-R4 with the V set of vectors x, where T(x)=0, is V...

A linear transformation from R3-R4 with the V set of vectors x, where T(x)=0, is V a subspace of R3?

*(4) (a) Prove that if p=(x,y) is in the set where y<x and if r=distance from...

*(4) (a) Prove that if p=(x,y) is in the set where y<x and if r=distance from p to the line y=x then the ball about p of radius r does not intersect with the line y=x. (b) Prove that the set where y<c is an open set. Justify your answer

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