Answer to Question #258073 in Discrete Mathematics for htd

Question

Let f be the function from x ={0, 1, 2, 3, 4, 5} to X defined by

f(x) = 4x mod 6.

Write f as a set of ordered pairs and draw the arrow diagram of f . Is f one-to-one? Is f onto?

Answer 1

It is given that $x=\{0,1,2,3,4,5\}.$

$$f(0)=4(0)\mod 6=0$$

$$f(1)=4(1)\mod 6=4$$

$$f(2)=4(2)\mod 6=2$$

$$f(3)=4(3)\mod 6=0$$

$$f(4)=4(4)\mod 6=4$$

$$f(5)=4(5)\mod 6=2$$

The given function as the set of ordered pairs

$$f = \{(0,0), (1,4), (2,2), (3,0), (4,4), (5,2)\}$$

Picture below represents arrow diagram of $f$

The function is called one-to-one if each $x$ from the domain has exactly one image in the range, and no $y$ in the range is image of more than one $x$ from the domain. As we can see, is the image of both and $3,$ so the function is not one-to-one.

The function is called onto if every $y$ in the range is the image of at least one $x$ from the domain. As we can see from diagram above, this function is onto.