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Introduction to Proofs/Contradiction
In this section, we will explore different techniques of proving a mathematical statement "If \(p\) then \(q\)". (\(p \to q\)).
Direct Proof
In this technique, we shall assume \(p\) and show that \(q\) is true.
Theorem \(\PageIndex{1}\)
Let \(n\) be an integer. If \(n\) is even then \(n^2\) is even.
- Proof
-
Assume that \(n\) is even. Then \(n=2m\) for some integer \(m \).
Consider \(n^2=(2m)^2=4m^2=2(2m^2).\) Since \( m \) is an integer, \( (2m^2)\) is an integer.
Thus \(n^2\) is even.
Example\(\PageIndex{1}\)
Show that for all integers \( n\), if \(n\) is odd then \(n^2\) is odd.
- Answer
-
Assume that \(n\) is odd. Then \(n=2m+1\) for some integer \(m \).
Consider \(n^2=(2m+1)^2=4m^2+4m+1=2(2m^2+2m)+1.\)
Since \( m \) is an integer, \( (2m^2+2m)\) is an integer.
Thus \(n^2\) is odd.
Proof by Contrapositive
In this technique, we shall assume \(\negp\) and show that \( \negq\) is true.
Theorem \(\PageIndex{2}\)
Let \(n\) be an integer. If \(n^2\) is even then \(n\) is even.
- Proof
-
We shall prove this statement by assuming \(n\) is odd. Then \(n=2m+1\) for some integer \(m \).
Consider \(n^2=(2m+1)^2=4m^2+4m+1=2(2m^2+2m)+1.\)
Since \( m \) is an integer, \( (2m^2)+2m\) is an integer.
Thus \(n^2\) is odd.
Example\(\PageIndex{2}\)
Show that for all integers \( n\), if \(n^2\) is odd then \(n\) is odd.
- Answer
-
We shall prove this statement by assuming \(n\) is even. Then \(n=2m\) for some integer \(m \).
Consider \(n^2=(2m)^2=4m^2=2(2m^2).\)
Since \( m \) is an integer, \( (2m^2)\) is an integer. Thus \(n^2\) is even.
Proof by Contradiction
In this technique, we shall assume the negation of the given statement is true, and come to a contradiction.
Theorem \(\PageIndex{3}\)
\(\sqrt{2}\) is irrational.
- Proof
-
Assume that \(\sqrt{2}\) is rational. Then \(\sqrt{2}= \dfrac {a}{b}\), where \(a \in \mathbb{Z}, b \in \mathbb{Z}\setminus \{0\}\), with no common factors between \(a\) and \(b\). Now, \( \sqrt{2} a=b\). Then \( 2a^2=b^2\). Since \(2\) divides \(2a^2\), \(2\) divides \(b^2\). Thus \(b^2\) is even. Therefore, \(b\) is even, (by theorem 2). Since \( b\) is even, \(2 \) divides \(b\). Therefore, \(2^2 \) divides \(b^2\).
Since \(2a^2=b^2\), \(2^2 \) divides \(2a^2\). Therefore, \(2 \) divides \(a^2\). Which implies \(a\) is even. This contradicts the fact that \(a\) and \(b\) have no common factors. Thus \(\sqrt{2}\) is irrational.
Proof by Counterexample
Example \(\PageIndex{3}\):
Decide whether the statement is true or false and justify your answer:
For all integers \(a,b,u,v\), and \(u\ne 0, v \ne 0\), if \(au+bv =0\) then \(a=b=0.\)
Solution: The statement is false.
Counterexample: Choose \(a=1,b=-1, u=2,v=2\), then \(au+bv =0\), but \(a\ne 0. b \ne 0, a \ne b.\)
Mathematical Induction
Process of Proof by Induction
Let \(p(n)\) be a mathematical statement, \(n \in \mathbb{N}\) i.e., \(n \ge1\).
-
Prove the statement is true for the lowest value of \(n\).
-
Assume that \(p(n)\) is true for all \(n=k\).
-
Prove \(p(k+1)\) is true.
Example \(\PageIndex{4}\)
Prove \(2^n>n+4\) for \(n\ge 3, n\in \mathbb{N}\).
- Answer
-
Let \(n=3\). Then \(2^3 >3+4\) is true since clearly \(8>7\). Thus the statement is true for \(n=3\).
Assume that \(2^n > n+4\) is true for some \(n=k\).
We will show that \(2^{k+1} > (k+1)+4\).
Consider \(2^{k+1}=2 \cdot 2^{k} >2 \cdot (k+4)=2k+8\).
Since \(2k > k+1\) and \(8 >4\), we have \(2k+8>(k+1)+4\).
Thus the statement is true for all \(n=k\).
By induction, \(2^n > n+4\) for all \(n\ge 3, n \in \mathbb{Z} \).◻
Example \(\PageIndex{5}\)
Show that \(9|(10^{n+1}+3(10^n)+5), \forall n \ge 1\).
- Answer
-
Let \(n=1\). Then \(9|(10^2)+3(10)+5\), which is \(9|135\), which is true since \(135=9(15)\) and \(15 \in \mathbb{Z}\).
Assume that \(9|(10^{n+1}+3(10^n)+5\) is true for some \(n=k\).
We will show that \(10^{k+1+1}+3(10^{k+1})+5=9m\) for some \(m \in \mathbb{Z}\).
Consider \(10^{k+1+1}+3(10^{k+1})+5=10(10^{k+1}+3(10^k)+5)-9(5)\)
\(=10(9m)-9(5)\)
\(=9(10m-5)\), where \(10m-5 \in \mathbb{Z}\).
By induction, \(9|(10^{n+1}+3(10^n)+5), \forall n \ge 1\).◻
Example \(\PageIndex{6}\)
Show that \(1+2+3+\cdots + n=\frac{n(n+1)}{2}, \; \forall \; n\ge 1\).
- Answer
-
Let \(n=1\). Then \(1=\frac{1(1+1)}{2}\) which is true.
Assume \(1+2+3+\cdots + n=\frac{n(n+1)}{2}\) is true for some \(n=k\).
We will show that \(1+2+3+\cdots + n +(n+1)=\frac{(n+1)(n+1+1)}{2}\)
Consider \(1+2+3+\cdots +n+(n+1)=[1+2+3+\cdots+n]+(n+1)\).
\(=\frac{n(n+1)}{2} +(n+1)\)
\(=\frac{n(n+1)+2(n+1)}{2}\)
\(=\frac{(n+1)(n+1+1)}{2}\).
By induction, \(1+2+3+\cdots + n=\frac{n(n+1)}{2}, \; \forall \; n\ge 1\).◻
Example \(\PageIndex{7}\)
Prove that \(3|(10^{n+1}+10^n+1), \; \forall \; n\ge 1\).
- Answer
-
Let \(n=1\). Then \(3|(10^2+10+1)\) is true since \(111=3(37)\) and \(37 \in \mathbb{Z}\).
Assume that \(3|(10^{n+1}+10^n+1)\) for some \(n=k\).
We will show that \(10^{k+1+1}+10^{k+1}+1=3m, m\in \mathbb{Z}\).
Consider \(10^{k+1+1}+10^{k+1}+1=10(10^{k+1}+10^k+1)-9(1)\)
\(=10(3m)-3(3)\)
\(=3(10m-3)\) where \(10m-3 \in \mathbb{Z}\).
By induction, \(3|(10^{n+1}+10^n+1), \; \forall \; n\ge 1\).◻
