The Nature of Proof

proof year12 ext2

The nature of proof | NSW Curriculum Websitey

• ME2-12-01 selects and applies the language, notation and methods of proof to prove results

📖 Prior Knowledge

Content	Prior knowledge	Used for
Probability	- set notation	- notation of proof
Further Work with Functions	- quadratic and cubic inequalities - absolute value	- proofs involving inequalities - proofs involving absolute value
Differential Calculus	- calculus	- proofs involving calculus
Proof by Mathematical Induction	- simple proofs involving MI	- further proofs involving MI

The language and notation of proof

• Use the formal language of proof, including the terms ‘statement’, ‘proposition’, ‘implication’, ‘converse’, ‘negation’, ‘contradiction’, ‘counterexample’, ‘equivalence’ and ‘contrapositive’
• Define a statement or proposition as a sentence that is either true or false, but not both
• Use the notation $P\wedge Q$ to represent the statement ’$P$ and $Q$’ and the notation $P\vee Q$ to represent the statement ’$P$ or $Q$’
• Define and use the negation of $P$ as ‘not $P$’, denoted $\neg P$ or $\sim P$
• Define an implication as an ‘if–then’ statement, where ‘if $P$ then $Q$’ is denoted $P\Rightarrow Q$ or $P\to Q$, read as ’$P$ implies $Q$’
• Use the quantifiers ‘for all’ $(\forall )$, and ‘there exists’ $(\exists )$ in formulating statements
• Negate statements including the negation of a negation $\neg (\neg P)=P$, the negation of an implication $\neg (P\Rightarrow Q)=(P\text{ and }\neg Q)=(P\wedge \neg Q)$, the negation $\neg (P\text{ and }Q)=(\neg P\text{ or }\neg Q)$, that is $\neg (P\wedge Q)=(\neg P\vee \neg Q)$, and the negation $\neg (P\text{ or }Q)=(\neg P\text{ and }\neg Q)$, that is $\neg (P\vee Q)=(\neg P\wedge \neg Q)$, noting that $P\Rightarrow Q=\neg (P\text{ and }\neg Q)=(\neg P\text{ or }Q)$, that is $P\Rightarrow Q=\neg (P\wedge \neg Q)=(\neg P\vee Q)$
• Define and use the converse of ‘if $P$ then $Q$’ as ‘if $Q$ then $P$’, denoted $Q\Rightarrow P$
• Recognise that the converse of a true implication may or may not be true
• Define equivalence of $P$ and $Q$, as both $P\Rightarrow Q$ and $Q\Rightarrow P$, denoted $P\iff Q$ or $P\leftrightarrow Q$, read as ’$P$ if and only if $Q$’, commonly abbreviated ’$P$ iff $Q$’
• Define the contrapositive of ‘if $P$ then $Q$’ as ‘if not $Q$ then not $P$’, denoted $\neg Q\Rightarrow \neg P$
• Recognise that an implication is equivalent to its contrapositive, that is $(P\Rightarrow Q)\iff (\neg Q\Rightarrow \neg P)$, and use this to prove results

Illustrations of proofs

• Use proof by contradiction to prove the truth of mathematical statements
• Use examples and counterexamples to test the truth of mathematical statements
• Prove results involving integers

Proof of inequalities

• Prove results involving inequalities using the definition of $a>b$ for real $a$ and $b$, that is $a>b$ if and only if $a-b>0$
• Prove results involving inequalities using the property that squares of real numbers are non-negative, in particular $(a\pm b{)}^2\ge 0$
• Prove and use results for numbers: if $a>b$ then $a\pm c>b\pm c$; if $a>b>0$ then $\frac{1}{b}>\frac{1}{a}>0$ and vice versa; if $|a|>|b|$ then $a^2>b^2$ and vice versa; if $a>b$ and $b>c$ then $a>c$; if $a>b$ and $c>d$ then $a+c>b+d$; if $a>b$ and $c>0$ then $ac>bc$; if $a>b$ and $c<0$ then $ac<bc$
• Prove and use the triangle inequality $|a+b|\le |a|+|b|$ and interpret the inequality geometrically
• Establish and use the relationship between the arithmetic mean and geometric mean for two non-negative numbers, that is $\forall a,b\ge 0,\frac{a+b}{2}\ge \sqrt{ab}$
• Prove results involving inequalities using previously obtained or known inequalities
• Prove inequalities involving geometry
• Prove results using the squeeze theorem: if $f(x)\le g(x)\le h(x)$ for all $x$ that are near $k$, but not necessarily at $k$, and $\underset{x\to k}{\lim }f(x)=\underset{x\to k}{\lim }h(x)=L$, then $\underset{x\to k}{\lim }g(x)=L$
• Prove inequalities using graphical or calculus techniques or a combination of both

Further proof by mathematical induction

• Prove results involving trigonometric, logarithmic, exponential, polynomial or other identities, including the binomial theorem, using mathematical induction
• Prove inequality results using mathematical induction
• Prove results in calculus using mathematical induction
• Explain that a recursive formula, or recurrence relation, is a formula that defines each term of a sequence using a preceding term
• Prove results involving first-order recursive formulas using mathematical induction
• Prove geometric results using mathematical induction