Question

Suppose that P(n) is a propositional function. Determine for which nonnegative integers n the statement P(n) must be true if a) P(0) is true; for all nonnegative integers n, if P(n) is true, then P(n 2) is true. b) P(0) is true; for all nonnegative integers n, if P(n) is true, then P(n 3) is true. c) P(0) and P(1) are true; for all nonnegative integers n, if P(n) and P(n 1) are true, then P(n 2) is true. d) P(0) is true; for all nonnegative integers n, if P(n) is true, then P(n 2) and P(n 3) are true

Answer 1

a).
`$P(0)$` is true

Then ,
`$P(0+2)=P(2)$` is true.

`$P(2+2)=P(4)$` is true

`$P(4+2)=P(6)$` is true.

Therefore, we see that
`$P(n)$` is true for all the even integers :
`$\{0, 2,4,6,...\}$`

b).
`$P(0)$` is true

Then ,
`$P(0+3)=P(3)$` is true.

`$P(3+3)=P(6)$` is true

`$P(6+3)=P(9)$` is true.

Therefore, we see that
`$P(n)$` is true for all the multiples of 3 :
`$\{0, 3,6,9,12,...\}$`

c).
`$P(0)$` and
`$P(1)$` is true, then
`$P(0+2)=P(2)$` is true

`$P(1)$` and
`$P(2)$` is true, then
`$P(1+2)=P(3)$` is true.

`$P(2)$` and
`$P(3)$` is true, then
`$P(2+2)=P(4)$` is true.

So, we observe that
`$P(n)$` is true for all the non- negative integers :
`$\{0, 1,2,3,4,5,6,...\}$`.

d).
`$P(0)$` is true,

So,
`$P(0+2)$` and
`$P(0+3)$` is true or
`$P(2)$` and
`$P(3)$` is true.

Now,
`$P(2)$` is true.

Again,
`$P(2+2)$` and
`$P(2+3)$` is true or
`$P(4)$` and
`$P(5)$` is true.

Now,
`$P(3)$` is true.

Again,
`$P(3+2)$` and
`$P(3+3)$` is true or
`$P(5)$` and
`$P(6)$` is true.

Thus,

`$P(n)$` is true for all the non- negative integers except 1 :
`$\{0, 2,3,4,5,6,...\}$`.