Question 11 Mark
State whether the following statement is true or false. Justify:
Let P(n) be a statement and let P(k) ⇒ P(k + 1), for some natural number k, then P(n) is true for all n ∈ N.
Let P(n) be a statement and let P(k) ⇒ P(k + 1), for some natural number k, then P(n) is true for all n ∈ N.
Answer
View full question & answer→False.Solution:
Given that, P(k) ⇒ P(k + 1) for some natural number k P(1) ⇒ P(2) but if $\text{P}(2)\nRightarrow\text{P}(3)$ [or $\text{P(k)}\nRightarrow\text{P(k+1)}$ for some k] Then P(n) will not be true for all n ∈ N. Hence, the statement is 'False'.
Given that, P(k) ⇒ P(k + 1) for some natural number k P(1) ⇒ P(2) but if $\text{P}(2)\nRightarrow\text{P}(3)$ [or $\text{P(k)}\nRightarrow\text{P(k+1)}$ for some k] Then P(n) will not be true for all n ∈ N. Hence, the statement is 'False'.