Discrete mathematics=> prove following question using universal specification, e

Question

Discrete mathematics=> prove following question using universal specification, existential specification, universal generalization and existential generalization.

1. For every natural number there exists another natural number which is greater than it.

Therefore, there exists no greatest natural number. (Hint : Proof by contradiction/Indirect proof)

Use N(x) to denote "x is natural number,"

G(x,y) to denote "x is greater than y"

T(x) to denote "x is a greatest natural number."

---------------------------------------------

Note that a greatest natural number is also a natural number. That is (All)x(T(x) -> N(x)). You can use it as a premise

Explanation / Answer

Proof by Contradiction

Let x is a natural number and is the greatest number.

so N(x) : TRUE

T(x) : TRUE.

Now consider another number x + 1

Since N(x) is TRUE, N(x+1) is also true.

x+1 is greater than x, so G(x+1, x) : TRUE

since x+1 is greater than x G(x+1,x),

so x can not be greatest number , T(x) can not be true.

so that proves there doesn;t exist any greatest number.

Related Questions

Navigate

• Browse (All)
• Browse D
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.