Proving Equivalence Relation: z~w $\in$ $\mathbb{C}$

In summary, to show that z~w is an equivalence relation, we need to show that z~z, z~y implies y~z, and z~y and y~x implies z~x. To prove the second condition, we can use the fact that if z/w is real, then its multiplicative inverse w/z is also real. This can be shown by multiplying through by the complex conjugate of y and dividing by the complex conjugate of z.
  • #1
latentcorpse
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0
I'm having a bit of difficulty in showing the following is an equivalence relation:

[itex]z~w iff \frac{z}{w} \in \mathbb{R} \forall z,w \in \mathbb{C}*=\mathbb{C} \backslash \{0\][/itex]

clearly z~z is ok as 1 is real

then i considered [itex]\frac{z}{y} \in \mathbb{R}[/itex] and tried to show y divided by z is real. i decided to multiply through by [itex]\frac{\bar{y}}{\bar{y}}[/itex] to get [itex]\frac{z \bar{y}}{|y|^2} \in \mathbb{R}[/itex]
then clearly [itex]z \bar{y} \in \mathbb{R}[/itex]
and dividing by [itex]|z|^2[/itex] we get [itex]\frac{\bar{y}}{\bar{z}} \in \mathbb{R}[/itex]
but now what?
 
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  • #2
1) If a number [tex]w[/tex] is real, what does that tell you about the inverse [tex]w^{-1}[/tex]? (remember: [tex]w * w^{-1} = 1[/tex])

2) For the final step, use the fact that you can write a fraction as: [tex]\frac{a}{b} = \frac{a}{c}\frac{c}{b}[/tex]
 
  • #3
(z/w)(w/z)= 1. Therefore, w/z is the multiplicative inverse of z/w. But z/w is real so its multiplicative inverse is ...
 
  • #4
erm.. just w/z?
 
  • #5
Yes, but what I was looking for is "since z/w is real, its multiplicative inverse, w/z, is real". That was what xepema meant.
 

What does it mean for two complex numbers to be equivalent under relation z~w?

Two complex numbers, z and w, are considered equivalent under relation z~w if and only if their real parts are equal and their imaginary parts are equal. In other words, z and w must be of the form a + bi, where a and b are real numbers, and a = c and b = d for them to be equivalent under z~w.

How do you prove that z~w is an equivalence relation?

To prove that z~w is an equivalence relation, we must show that it satisfies three properties: reflexivity, symmetry, and transitivity. Reflexivity means that every complex number is equivalent to itself, which is satisfied by z~w since a = a and b = b. Symmetry means that if z~w, then w~z, which can be easily shown by the fact that if a = c and b = d, then c = a and d = b. Lastly, transitivity means that if z~w and w~x, then z~x, which can be proven by substituting the equivalent values for z~w and w~x into the definition of z~x.

Can you give an example of two complex numbers that are equivalent under z~w?

One example of two complex numbers that are equivalent under z~w is 2 + 3i and 2 + 3i. Since their real parts are both 2 and their imaginary parts are both 3, they satisfy the definition of z~w and are therefore equivalent.

What is the significance of proving equivalence relation in mathematics?

Proving equivalence relation is important in mathematics as it helps us categorize and understand different mathematical concepts. It allows us to group together objects that share certain properties and treat them as equivalent, making it easier to analyze and manipulate them in mathematical expressions. This concept is also fundamental in fields such as abstract algebra and topology, where equivalence relations are used to define and study more complex structures.

Are there other types of equivalence relations besides z~w?

Yes, there are many other types of equivalence relations in mathematics. Some common examples include numerical equivalence, where two numbers are considered equivalent if they have the same numerical value, and congruence, where two geometric figures are considered equivalent if they have the same shape and size. In general, an equivalence relation can be defined for any set of objects, as long as it satisfies the three properties of reflexivity, symmetry, and transitivity.

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