Existential Instantiation Proof Problem: Absolute Value

Subsection 5.2.3 Existential Instantiation Proof Problem: Absolute Value

Absolute Values

Suppose our universe is the set of integers and we assume that every integer has both an absolute value and another number that is its successor. Thus every integer has an absolute value.

Assign the following names to basic statements:

A(x): True if x has an absolute value.

S(x): True if x has a successor.

Prove: ∀x (∃y (A(x) ∧ S(y))) Every integer has an absolute value and a successor.

∴ ∀z (A(z)) Every integer has an absolute value.

You should do this proof yourself. (Hint: Be careful. Notice that the existentially quantified variable occurs within the scope of a universally quantified one. You will need to instantiate y as a function of the variable you used when you instantiated x.)

You can also watch our video, which will outline our strategy for doing this.

Exercises Exercises

1.

Assume the following premise:

[1] ∃x (∀y (R(x, y))) Premise

Prove: ∀y (∃x (R(x, y)))

Answer.

questionId: PsQs02 problemType: gradeLogicProof questionTitle: Instantiation and Generalization Practice questionDisplayText: Prove an interesting general claim about swapping quantifiers.. hints: Instantiate both quantifiers in the premise. Remember to go outside in. [1] x (y (R(x, y))) Premise [2] y (R(a*, y)) Existential Instantiation [1] [3] R(a*, b) Universal Instantiation [2] [4] x (R(x, b)) Existential Generalization [3] [5] y (x (R(x, y))) Universal Generalization [4]

Solution.