MAT102 Tutorial Quiz 2

• Define a function $f:A\to B$ and a relation on $A$ as $\sim$, where $a\sim b$ means $f(a)=f(b)$
• Show that if $\sim$ is antisymmetric if and only if $f$ is injective.
• If $a\sim b$ and $b\sim a$ then $a=b$
• $⟹$
  • Assume it's antisymmetric
  • So $a\sim b⟹f(a)=f(b)$
  • By assumption, $f(a)=f(b)⟹a=b$.
  • So it's injective
• $⟸$
  • Assume it's injective
  • So $f(a)=f(b)⟹a=b$
  • So then $a\sim b$ and $b\sim a$ means the same thing essentially.
  • $f(a)=f(b)$ or $f(b)=f(a)$, either way, still antisymmetric