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0.5: Proof Templates

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  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Proof Templates

    • To prove injective relationship (ie prove one to one)
      Is \(f(x_1) = f(x_2)\)?

    YES then you need to prove \(x_1=x_2\).

    Let \(A\) be a set.   Let \(x_1,x_2 \in A\) such that \(f(x_1)=f(x_2)\).


    Show \(x_1=x_2\).

    NO then you need to provide one counterexample where \(x_1 \ne x_2\) when \(f(x_1) = f(x_2)\).

    • Is there a surjective relationship (i.e. prove onto)?

    YES Note we are starting from \(f: A \rightarrow B\). Thus \(f(A)\subseteq B\) by definition.

    Let \(A\) and \(B\) be sets where \(A\) is onto \(B\) if and only if \(f(A) \subseteq B\) and \(B \subseteq f(A)\).

    By definition, \(f(A) \subseteq B\).  Next show \(B \subseteq f(A)\).

    Let \(x \in B\).


    Show \(x \in f(A)\).

    Conclude with:  Since \(f(A) \subseteq B\) and \(B \subseteq f(A)\)  therefore \(f\) is onto.◻

    NO then need to find a counterexample where \(b \in B\), but there is no \(a \in A\) whereby \(f(a)=b\).

    • Reflexive, Symmetric, Anti Symmetric, Transitive Property Proofs 

    Need a non empty set \(A\) and a relation \(R\).

    • Reflexive

    Let \(a \in A\).  ← Starting point


    Show \(aRa\) ← Ending point

    • Symmetric

    Let \(a, b \in A\) s.t. \(aRb\).


    Show \(bRa\)

    • Antisymmetric

    Let \(a,b \in A\) s.t. \(aRb\) and \(bRa\)


    Show \(a=b\).

    • Transitive

    Let \(a,b,c \in A\) s.t. \(aRb\) and \(bRa\).


    Show \(aRc\).

    0.5: Proof Templates is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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