Question

Problem ( Proving theorems!)

For each of the following statements, translate it into predicate logic and prove it, if the statement is true, or disprove it, otherwise:

1. 1.

   for any positive integer, there exists a second positive the square of which is equal to the first integer,

2. 2.

   for any positive integer, there exists a second positive integer which is greater or equal to the square of the the first integer,

3. 3.

   for any positive integer, there exists a second positive which is greater or equal to the square of the the first integer, but smaller than the cube of the the first integer.

Answer 1

If a statement is true, we need to prove it in generalised way. But if a statement is false, one point/example is enough to prove it.

1: Statement 1 is FALSE.
Statement:  For any positive integer (say, 1,2,3,4). there exist a second positive integer whose square is equal to 1st integer.
This fails for I₁ = positive integer which is not perfect square (eg, 2, 3, 5, 6, 7, ...).

2: Statement 2 is TRUE
Statement: For any positive integer, there exists a second positive integer which is greater or equal to the square of the the first integer.
Square of 1st integer is also an integer. Since integer goes till INFINITY, hence, for all N from positive integer set, there exist an N+1 in positive integer set..

3: Statement 3 is FALSE
Statement: For any positive integer, there exists a second positive which is greater or equal to the square of the the first integer, but smaller than the cube of the the first integer.
Positive integers are {1,2,3,4,5,...}
The statement holds for N > 1 in positive Integer set. But for N=1 in the set, square of 1 is 1 and cube is also 1.
According to statement, N₁² <= N₂ < N₁³
But 1<=1<1 doesnot hold true for 2nd inequality.

Hope it helps.