Chapter 5 - Proof by induction
Questions with Correct Answers
principle of mathematical induction - ANSWERSif the base case (for n = 1) is true and
inductive step is true, then the theorem holds for all positive integers
Let S(n) be a statement parameterized by a positive integer n. Then S(n) is true for all
positive integers n, if:
1. Base case: S(1) is true .
2. Inductive step: For all k ∈ Z+, S(k) implies S(k+1)
base case - ANSWERSestablishes that the theorem is true for the first value in the
sequence
inductive step - ANSWERSestablishes that if the theorem is true for k, then the theorem
also holds for k + 1
inductive hypothesis - ANSWERSIn the statement "S(k) implies S(k+1)" of the inductive
step, the supposition that S(k) is true
Structural induction - ANSWERSa type of induction used to prove theorems about
recursively defined sets that follows the structure of the recursive definition
Show that for every x in a recursively defined set P, x has a particular property -
ANSWERSBase case: Show that every element in the basis satisfies the property.
Inductive step:
The recursive rules show how to construct larger elements in P using smaller elements
from P.
Assume the property holds for the smaller elements and prove that the property holds
for the larger elements.
Show that for every properly nested set of parentheses (every x in P), x is balanced
(left[x] = right[x]) - ANSWERSBasis: Show that () is balanced
Inductive step:
Case 1: Assume u is balanced, Prove that (u) is balanced.
