Really confused on how to put an input with 2 outputs in set builder notation. Please Help if possible. Like would it be {X | _ < X <_} or {_, _)}

Question

How do I use Set Builder Notation when the input has 2 outputs?

For example; 4z(z+5) - 1(z+5) = 0​ 4z^2 + 20z – z –5 = 0​ 4z^2 + 19z – 5 = 0​

Plugging in

​ A = 4​

B = 19

C = -5​ ​

-19 ± √19^2 – 4 x 4(-5)/2 x 4​ ​ -19 + 21/8 = 2/8 = 1/4​ -19 – 21/8 = -40/8 = -5​ ​

Z is equal to 1/4 and -5

How would Z be put in a Set Notation? Would it be { Z | -5 ≤ Z ≥ 1/4 } ? Or would it be {-5, 1/4}? I'm really confused please help if possible!

Answer 1

You're on the right track but you need to use equal signs (=) rather ≤ or ≥. Also, you should include the number set that z is in. If it's real numbers use R; if it's complex numbers, use C:

{ z∈R | (z = -5) or (z = 1/4) }

Read as "all z in the set of real numbers such that z equals -5 or z equals 1/4." Sometimes you can use the symbol U, which is the union operator, to represent the or. However, use of the word "or" is common.

Answer 2

{ Z | -5 ≤ Z ≤ 1/4 } means that Z can be any value greater than -5 and less than 1/4

(So, Z could be -4, -2.5, 1/8, etc.)

{-5, 1/4} is the set that contains only -5 and 1/4.

For this application, it looks like {-5, 1/4} is what you are looking for.