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Activities 7.4 Activities
Below is a more detailed version of
Procedure 7.1.2 . Follow the steps of
Procedure 7.4.1 to create a proof by induction for each of the requested proofs in this activity set.
Procedure 7.4.1 . Mathematical induction, step-by-step.
Write the statement with \(n\) replaced by \(k\text{.}\)
Write the statement with \(n\) replaced by \(k+1\text{.}\)
Identify the connection between the \(\nth[k]\) statement and the \(\nth[(k+1)]\) statement.
Complete the induction step by assuming that the \(n = k\) version of the statement is true , and using this assumption to prove that the \(n = k + 1\) version of the statement is true .
Complete the induction proof by proving the base case .
Activity 7.1 .
A binary string is a “word” in which each “letter” can only be \(0\) or \(1\text{.}\)
Prove that there are \(2^n\) different binary strings of length \(n\text{.}\)
Activity 7.2 .
Prove that for every positive integer \(n\text{,}\) the binomial \(1-x^n\) can be factored as \((1-x)(1+x+x^2+\dotsb + x^{n-1})\text{.}\)
Activity 7.3 .
Prove that the following argument is valid for all positive integers \(n\text{.}\)
\((p_1 \lgcand q_1) \lgccond r_1\) \((p_2 \lgcand q_2) \lgccond r_2\) \(\phantom{(p_2 \lgcand q_2)} \vdots \) \((p_n \lgcand q_n) \lgccond r_n\) \(p_1 \lgcand p_2 \lgcand \dotsb \lgcand p_n\) \((q_1 \lgccond r_1) \lgcand (q_2 \lgccond r_2) \lgcand \dotsb \lgcand (q_n \lgccond r_n)\)
Careful. Recall that in this context, the words valid and true do not have the same meaning.
Activity 7.4 .
Prove that a truth table involving \(n\) statement variables requires \(2^n\) rows.
Activity 7.5 .
Prove that a knight can be moved from any square to any other square on an \(n \times n\) chess board by some sequence of allowed moves, for every \(n\ge 4\text{.}\)
