# 0.5: Proof Templates

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### 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.