Types of Proof 1. Deductive Proof
Start with known facts and deduce what you are trying to prove.
2. Inductive Proof
Assume what you are trying to prove and induce a solution.
Types of Proof 1. Deductive Proof
Start with known facts and deduce what you are trying to prove.
2. Inductive Proof
Assume what you are trying to prove and induce a solution.
3. Proof by Contradiction
Assume the opposite of what you are trying to prove and create a
contradiction.
Contradiction means the original assumption is incorrect, therefore
the opposite must be true.
Inequality Techniques 0 prove easier to becan it , prove To yxyx
2
Prove 1995 e.g.22 qp
pqi
pqqp
2
22
Inequality Techniques 0 prove easier to becan it , prove To yxyx
2
Prove 1995 e.g.22 qp
pqi
pqqp
2
22
2
2 22 qpqp
Inequality Techniques 0 prove easier to becan it , prove To yxyx
2
Prove 1995 e.g.22 qp
pqi
pqqp
2
22
2
2 22 qpqp
2
2qp
0
Inequality Techniques 0 prove easier to becan it , prove To yxyx
2
Prove 1995 e.g.22 qp
pqi
pqqp
2
22
2
2 22 qpqp
2
2qp
0
pqqp
2
22
Inequality Techniques 0 prove easier to becan it , prove To yxyx
2
Prove 1995 e.g.22 qp
pqi
pqqp
2
22
2
2 22 qpqp
2
2qp
0
pqqp
2
22
OR 2 2
Assume 2
p qpq
Inequality Techniques 0 prove easier to becan it , prove To yxyx
2
Prove 1995 e.g.22 qp
pqi
pqqp
2
22
2
2 22 qpqp
2
2qp
0
pqqp
2
22
OR 2 2
Assume 2
p qpq
2 22pq p q
2 2
2
0 2
0 ( )
p pq q
p q
Inequality Techniques 0 prove easier to becan it , prove To yxyx
2
Prove 1995 e.g.22 qp
pqi
pqqp
2
22
2
2 22 qpqp
2
2qp
0
pqqp
2
22
OR 2 2
Assume 2
p qpq
2 22pq p q
2 2
2
0 2
0 ( )
p pq q
p q
2But ( ) 0p q
Inequality Techniques 0 prove easier to becan it , prove To yxyx
2
Prove 1995 e.g.22 qp
pqi
pqqp
2
22
2
2 22 qpqp
2
2qp
0
pqqp
2
22
OR 2 2
Assume 2
p qpq
2 22pq p q
2 2
2
0 2
0 ( )
p pq q
p q
2But ( ) 0p q 2 2
2
p qpq
acbcabcbaii 222 Prove a) 1994
02ba
02 22 baba
abba 222
bccb
acca
2
2
22
22
bcacabcba 222222 222
bcacabcba 222
acbcabcbaii 222 Prove a) 1994
02ba
02 22 baba
abba 222
bccb
acca
2
2
22
22
bcacabcba 222222 222
bcacabcba 222
3
1 prove ,1 If b) bcacabcba
acbcabcbaii 222 Prove a) 1994
02ba
02 22 baba
abba 222
bccb
acca
2
2
22
22
bcacabcba 222222 222
bcacabcba 222
3
1 prove ,1 If b) bcacabcba
bcacabcbacba 22222
acbcabcbaii 222 Prove a) 1994
02ba
02 22 baba
abba 222
bccb
acca
2
2
22
22
bcacabcba 222222 222
bcacabcba 222
3
1 prove ,1 If b) bcacabcba
bcacabcbacba 22222
bcacabbcacabcba 22
acbcabcbaii 222 Prove a) 1994
02ba
02 22 baba
abba 222
bccb
acca
2
2
22
22
bcacabcba 222222 222
bcacabcba 222
3
1 prove ,1 If b) bcacabcba
bcacabcbacba 22222
bcacabbcacabcba 22
23 cbabcacab
acbcabcbaii 222 Prove a) 1994
02ba
02 22 baba
abba 222
bccb
acca
2
2
22
22
bcacabcba 222222 222
bcacabcba 222
3
1 prove ,1 If b) bcacabcba
bcacabcbacba 22222
bcacabbcacabcba 22
23 cbabcacab
3
1
13
bcacab
bcacab
3
3
1 Prove c) abccba
bcacabcba 222
0222 bcacabcba
0222 bcacabcbacba
022322
2223222223
bcacabcccbca
cbabcabbcbbaabccabaacaba
3
3
1 Prove c) abccba
bcacabcba 222
0222 bcacabcba
0222 bcacabcbacba
022322
2223222223
bcacabcccbca
cbabcabbcbbaabccabaacaba
03333 abccba
3
3
1 Prove c) abccba
bcacabcba 222
0222 bcacabcba
0222 bcacabcbacba
022322
2223222223
bcacabcccbca
cbabcabbcbbaabccabaacaba
03333 abccba
abccba 333
3
1
3
3
1 Prove c) abccba
bcacabcba 222
0222 bcacabcba
0222 bcacabcbacba
022322
2223222223
bcacabcccbca
cbabcabbcbbaabccabaacaba
03333 abccba
abccba 333
3
1
3
1
3
1
3
1
,,let ccbbaa
3
3
1 Prove c) abccba
bcacabcba 222
0222 bcacabcba
0222 bcacabcbacba
022322
2223222223
bcacabcccbca
cbabcabbcbbaabccabaacaba
03333 abccba
abccba 333
3
1
3
1
3
1
3
1
,,let ccbbaa
3
1
3
1
3
1
3
1cbacba
3
3
1 Prove c) abccba
bcacabcba 222
0222 bcacabcba
0222 bcacabcbacba
022322
2223222223
bcacabcccbca
cbabcabbcbbaabccabaacaba
03333 abccba
abccba 333
3
1
3
1
3
1
3
1
,,let ccbbaa
3
1
3
1
3
1
3
1cbacba
3
3
1abccba
nn
n aaan
aaa
21
21
Mean GeometricMean Arithmetic
1 prove ,8111 Suppose d) xyzzyx
81
8111
xyzyzxzzxyyx
zyx
nn
n aaan
aaa
21
21
Mean GeometricMean Arithmetic
1 prove ,8111 Suppose d) xyzzyx
81
8111
xyzyzxzzxyyx
zyx
3
3
1xyzzyx AM GM
nn
n aaan
aaa
21
21
Mean GeometricMean Arithmetic
1 prove ,8111 Suppose d) xyzzyx
81
8111
xyzyzxzzxyyx
zyx
3
3
1xyzzyx
33 xyzzyx
AM GM
nn
n aaan
aaa
21
21
Mean GeometricMean Arithmetic
1 prove ,8111 Suppose d) xyzzyx
81
8111
xyzyzxzzxyyx
zyx
3
3
1xyzzyx
33 xyzzyx
33 xzyzxyxzyzxy
AM GM
nn
n aaan
aaa
21
21
Mean GeometricMean Arithmetic
1 prove ,8111 Suppose d) xyzzyx
81
8111
xyzyzxzzxyyx
zyx
3
3
1xyzzyx
33 xyzzyx
33 xzyzxyxzyzxy
3 2223 zyxxzyzxy
AM GM
nn
n aaan
aaa
21
21
Mean GeometricMean Arithmetic
1 prove ,8111 Suppose d) xyzzyx
81
8111
xyzyzxzzxyyx
zyx
3
3
1xyzzyx
33 xyzzyx
33 xzyzxyxzyzxy
3 2223 zyxxzyzxy
233 xyzxzyzxy
AM GM
9 2 2 2 1 1 1
Prove iiia b c a b b c a c a b c
1 1 1
2 2
4
a b aba b ab
1 1 4
a b a b
1 1 4
1 1 4
b c b c
a c a c
9 2 2 2 1 1 1
Prove iiia b c a b b c a c a b c
1 1 1
2 2
4
a b aba b ab
1 1 4
a b a b
1 1 4
1 1 4
b c b c
a c a c
2 2 2 4 4 4
a b c a b b c a c
9 2 2 2 1 1 1
Prove iiia b c a b b c a c a b c
1 1 1
2 2
4
a b aba b ab
1 1 4
a b a b
1 1 4
1 1 4
b c b c
a c a c
2 2 2 4 4 4
a b c a b b c a c
1 1 1 2 2 2
a b c a b b c a c
9 2 2 2 1 1 1
Prove iiia b c a b b c a c a b c
1 1 1
2 2
4
a b aba b ab
1 1 4
a b a b
1 1 4
1 1 4
b c b c
a c a c
2 2 2 4 4 4
a b c a b b c a c
1 1 1 2 2 2
a b c a b b c a c
3 31 1 1 1
3 3
9
a b c abca b c abc
9 2 2 2 1 1 1
Prove iiia b c a b b c a c a b c
1 1 1
2 2
4
a b aba b ab
1 1 4
a b a b
1 1 4
1 1 4
b c b c
a c a c
2 2 2 4 4 4
a b c a b b c a c
1 1 1 2 2 2
a b c a b b c a c
3 31 1 1 1
3 3
9
a b c abca b c abc
1 1 1 9
a b c a b c
9 2 2 2 1 1 1
Prove iiia b c a b b c a c a b c
1 1 1
2 2
4
a b aba b ab
1 1 4
a b a b
1 1 4
1 1 4
b c b c
a c a c
2 2 2 4 4 4
a b c a b b c a c
1 1 1 2 2 2
a b c a b b c a c
3 31 1 1 1
3 3
9
a b c abca b c abc
1 1 1 9
a b c a b c
1 1 1 9
2
2 2 2 9
a b b c a c a b c
a b b c a c a b c
9 2 2 2 1 1 1
Prove iiia b c a b b c a c a b c
1 1 1
2 2
4
a b aba b ab
1 1 4
a b a b
1 1 4
1 1 4
b c b c
a c a c
2 2 2 4 4 4
a b c a b b c a c
1 1 1 2 2 2
a b c a b b c a c
3 31 1 1 1
3 3
9
a b c abca b c abc
1 1 1 9
a b c a b c
1 1 1 9
2
2 2 2 9
a b b c a c a b c
a b b c a c a b c
9 2 2 2 1 1 1
a b c a b b c a c a b c
9 2 2 2 1 1 1
Prove iiia b c a b b c a c a b c
1 1 1
2 2
4
a b aba b ab
1 1 4
a b a b
1 1 4
1 1 4
b c b c
a c a c
2 2 2 4 4 4
a b c a b b c a c
1 1 1 2 2 2
a b c a b b c a c
3 31 1 1 1
3 3
9
a b c abca b c abc
1 1 1 9
a b c a b c
1 1 1 9
2
2 2 2 9
a b b c a c a b c
a b b c a c a b c
9 2 2 2 1 1 1
a b c a b b c a c a b c
Inequalities Sheet
Exercise 10D
Note: Cambridge 8H (Book 1); 28
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