Find the truth set in case of each of the following open sentences defined on N:
(i) x + 2 < 10

(ii) x + 5 < 4

(iii) x + 3 > 2

Question

Find the truth set in case of each of the following open sentences defined on N:(i) x + 2 < 10(ii) x + 5 < 4(iii) x + 3 > 2

Philoid · Accepted Answer

The open sentence x + 2 < 10 is defined on N; the set of natural numbers.

N: {1, 2, 3, 4…}

x = 1 → x + 2 = 3 < 10

x = 2 → x + 2 = 4 < 10

x = 3 → x + 2 = 5 < 10

x = 4 → x + 2 = 6 < 10

x = 5 → x + 2 = 7 < 10

x = 6 → x + 2 = 8 < 10

x = 7 → x + 2 = 9 < 10

x = 8 → x + 2 = 10

So, ∃ x ∈ N, such that x + 2 < 10

x = {1, 2, 3, 4, 5, 6, 7} satisfies x + 2 <10.

So, the truth set of open sentence x + 2 < 10 defined on N is,

{1, 2, 3, 4, 5, 6, 7}

(ii) The open sentence x + 5 < 4 is defined on N; the set of natural numbers.

N: {1, 2, 3, 4…}

x = 1 → 1 + 5 = 6 > 4

So, the truth set of open sentence x + 5 < 4 defined on N is an empty set, {}.

(iii) The open sentence x + 3 > 2 is defined on N; the set of natural numbers.

N: {1, 2, 3, 4…}

x = 1 → x + 3 = 4 > 2

x = 2 → x + 3 = 5 > 2

x = 3 → x + 3 = 6 > 2

x = 4 → x + 3 = 7 > 2

x = 5 → x + 3 = 8 > 2

x = 6 → x + 3 = 9 > 2

And so on...

So, ∃ x ∈ N, such that x + 3 > 2

x = {1, 2, 3, 4, 5, 6, 7….} satisfies x + 3 > 2.

So, the truth set of open sentence x + 3 > 2 defined on N is an infinite set as there is infinite natural numbers satisfying the equation x + 3 > 2.

{1, 2, 3, 4, 5, 6, 7….}