#Proof #Year12 #Ext2 >[!info]- [The nature of proof | NSW Curriculum Websitey](https://curriculum.nsw.edu.au/learning-areas/mathematics/mathematics-extension-2-11-12-2024/content/n12/faabcfa8e0) >- ME2-12-01 selects and applies the language, notation and methods of proof to prove results ## 📖 Prior Knowledge | Content | Prior knowledge | Used for | | ------------------------------------- | ------------------------------------------------------ | -------------------------------------------------------------------- | | [[Probability and Data\|Probability]] | - set notation | - notation of proof | | [[Further Work with Functions]] | - quadratic and cubic inequalities<br>- absolute value | - proofs involving inequalities<br>- 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 $P \land Q$ to represent the statement '$P$ and $Q and the notation $P \lor Q$ to represent the statement '$P$ or $Q - Define and use the negation of $P$ as 'not $P, denoted $\lnot P$ or $\sim P$ - Define an implication as an 'if–then' statement, where 'if $P$ then $Q is denoted $P \Rightarrow Q$ or $P \rightarrow Q$, 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 $\lnot(\lnot P)=P$, the negation of an implication $\lnot(P \Rightarrow Q)=(P \text{ and } \lnot Q)=(P \land \lnot Q)$, the negation $\lnot(P \text{ and } Q)=(\lnot P \text{ or } \lnot Q)$, that is $\lnot(P \land Q)=(\lnot P \lor \lnot Q)$, and the negation $\lnot(P \text{ or } Q)=(\lnot P \text{ and } \lnot Q)$, that is $\lnot(P \lor Q)=(\lnot P \land \lnot Q)$, noting that $P \Rightarrow Q=\lnot(P \text{ and } \lnot Q)=(\lnot P \text{ or } Q)$, that is $P \Rightarrow Q=\lnot(P \land \lnot Q)=(\lnot P \lor Q)$ - 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 $P \Leftrightarrow Q$ 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 $\lnot Q \Rightarrow \lnot P$ - Recognise that an implication is equivalent to its contrapositive, that is $(P \Rightarrow Q) \Leftrightarrow (\lnot Q \Rightarrow \lnot P)$, 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 $(a \pm b)^2 \geq 0$ - 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 $|a + b| \leq |a| + |b|$ and interpret the inequality geometrically - Establish and use the relationship between the arithmetic mean and geometric mean for two non-negative numbers, that is $\forall a,b \geq 0, \frac{a + b}{2} \geq \sqrt{ab}$ - Prove results involving inequalities using previously obtained or known inequalities - Prove inequalities involving geometry - Prove results using the squeeze theorem: if $f(x) \leq g(x) \leq h(x)$ for all $x$ that are near $k$, but not necessarily at $k$, and $\lim_{x \to k} f(x) = \lim_{x \to k} h(x) = L$, then $\lim_{x \to k} g(x) = L$ - 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