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Readability settings Prev Up Next \(\require{cancel}
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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
\(n\ge 4 \text{.}\)
