Question

If $X=\{1,3,5,7\}$, then which of the following is a function from $X$ to itself?

• A. $f_1=\left\{\right(x,y):y=x+2\}$
• B. $f_2=\left\{\right(x,y):x+y<8\}$
• C. $f_3=\left\{\right(x,y):x+y=8\}$
• D. $f_4=\left\{\right(x,y):x<y\}$

Answer 1

The correct option is C $f_3=\left\{\right(x,y):x+y=8\}$

$f_1=\left\{\right(x,y):y=x+2\}$
elements of
$f_1=(1,3)(3,5)(5,7)(7,9)$
$g$ doesn't belongs to codomain $f_1$ so it is not a function to itself.

$f_2=\left\{\right(x,y):x+y<8\}$
elements of
$f_2=(1,3)(1,5)(3,1)(5,1)$
since $1$ has multiple images it is not a function.

$f_3=\left\{\right(x,y):x+y=8\}$
elements of $f_3=(1,7)(3,5)(5,3)(7,1)$
each element has unique image and within its codomain so it is function to itself

$f_4=\left\{\right(x,y):x<y\}$
elements of $f_4=(1,3)(1,5)(1,7)(3,5)(3,7)(5,7)$
since its elements have multiple images it is not a function