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  • 2025-03-17

MAT102 Prep 10

• Functions in Math
• Exercise 3
  • $A$ and $B$ are sets $A\subseteq B$.
  • $g$ is a relation on $A\times B$
  • $S_g=A\times A\subseteq A\times B$.
  • Does $g$ define a function?
  • We need to check left-total and vertical line test (functionality).
  • Left-total
  • This means for every $a\in A$ there is a $b\in B$ such that $agb$
    • We don't know the relation on $A\times B$, we just know that it's something.
    • But it must produce something thats in $A\times B$. $A\times A\subseteq A\times B$ so anything produced by $A\times A$ will be in $A\times B$.
    • But we can't know if it's going to be in $B$ specifically.
  • As for functionality,
    • This is the vertical line test.
    • $A\times A$'s results must only appear once in $A\times B$.
  • Example:
    • $A=\{0,1\}$
    • $B=\{0,1,2\}$
    • $A\times A=\{(0,1),(1,1),(1,0),(0,0)\}$
    • $A\times B=\{\begin{array}{c}(0,0),(0,1),(0,2)\\ (1,0),(1,1),(1,2)\\ (2,0),(2,1),(2,2)\end{array}\}$
• Exercise 4
  • Suppose that $X$ is an arbitrary set and $\mathrm{\varnothing }$ is the empty set. Is there a function $f:\mathrm{\varnothing }\to X$? Is there $g:X\to \mathrm{\varnothing }$
  • $\mathrm{\varnothing }\times X=\mathrm{\varnothing }$
    • We have $(a,b)\in S$
    • $a$ is from $\mathrm{\varnothing }$
    • $b$ is from $X$
    • Every $a$ must appear in $b$.
    • This is true, nothing can appear in $X$. This is because we don't have to check, because there's nothing to check.
    • Subset:
      • let $a\in \mathrm{\varnothing }$
      • $a\in X$
      • By definition, $\mathrm{\varnothing }\subseteq X$
    • Functionality:
      • The $\mathrm{\varnothing }$ only appears in $X$ once.
    • Because we have left-total and functionality, there is a function that exists for $f$
  • $X\times \mathrm{\varnothing }=\mathrm{\varnothing }$
    • So here $X$ can't ever produce something in the $\mathrm{\varnothing }$.
    • But we have $a\in X$ and $b\in \mathrm{\varnothing }$
    • Only if $X=\mathrm{\varnothing }$ then can the elements $a$ be in $\mathrm{\varnothing }$.
    • aka $\text{nothing}\in \mathrm{\varnothing }$.
    • Left-totality:
      • Let $a\in X$
      • By definition $a\notin \mathrm{\varnothing }$
      • So it cannot be in the $\mathrm{\varnothing }$
    • Functionality:
      • Let $a\in X$
      • …
      • By the same argument as above, it doesn't even appear once.
      • So it is not functional.
    • Since the function $g$ is not left-total or functional, there is no function such that $g:X\to \mathrm{\varnothing }$
• Exercise 5
  • Which are equal to one another?:
  • $f:\{0,1\}\to \mathbb{R},x\mapsto x$
    • $f(x)=x$
    • Input: $\{0,1\}$
    • Output: $\mathbb{R}$
    • Test
    • $f(0)=0$
    • $f(1)=1$
  • $g:\{0,1\}\to \mathbb{R},x\mapsto x^2$
    • $g(x)=x^2$
    • Input: $\{0,1\}$
    • Output: $\mathbb{R}$
    • Test
    • $g(0)=0^2=0$
    • $g(1)=1^2=1$
  • $h:\{0,1\}\to \{0,1\},x\mapsto x$
    • $h(x)=x$
    • Input: $\{0,1\}$
    • Output: $\{0,1\}$
    • Test
    • $h(0)=0$
    • $h(1)=1$
  • The outputs are all equal, but the Domain and Codomain must be the same.
  • So only $f=g$.
• Images and Preimages

• 1
  • $f:\mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}$
  • $f(n,m)=2n-m$
  • $2(10)-(-4)$
  • $24$
• 2
  • $f_2\left(\frac{a}{b}\right)=a$
  • Only if $b=1$

• 3