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CS 1050 B: Construction Proofs
January 31, 2008
Solutions to Homework 2
Lecturer: Sasha Boldyreva
Problem 2.1, 5 points.
Use mathematical induction to prove that 2
n
+ 3
≤
2
n
for all
n
≥
4.
Base case. n=4: 11
≤
16  True.
Inductive step.
Inductive hypothesis: for all
n
≥
4
,
2
n
+ 3
≤
2
n
.
To prove: for all
n
≥
4
,
2(
n
+ 1) + 3
≤
2
n
+1
.
2(
n
+ 1) + 3 = (2
n
+ 3) + 2
≤
2
n
+ 2
≤
2
·
2
n
= 2
n
+1
The ﬁrst inequality used the inductive hypothesis, the second used the fact that 2
≤
2
n
for
n
≥
1.
Problem 2.2, 5 points.
Use mathematical induction to show that
n
distinct lines in
the plane passing through the same point divide the plane into 2
n
regions.
Base case. n=1. One lined divides the plane into 2 regions. Inductive step. Inductive
hypothesis: For
n >
0 n lines divide the plane into 2n regions. To show: For
n >
0 n+1 lines
divide the plane into 2(n+1) regions. Look at
n
lines ﬁrst. By the inductive hypothesis,
they divide the plane into 2
n
regions. Whenever an extra line added, it divides two existing
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