1

I have been trying to understand the differences between the symbols “$=$”, “$\equiv$” and “$:=$” through the use of Wikipedia. While I can see that “$=$” is different from the other two, I am not really seeing the difference between $\equiv$ and $:=$. According to Wikipedia, the former indicates identity, whereas the latter indicates definition. I have been trying to figure out the difference between an identity and a definition, but I really didn’t succeed.

Could anybody help me see the difference?

EoDmnFOr3q
  • 1,356
  • 2
    This question is probably a duplicate, but a definition tells you the meaning of a symbol or a word. So, for example, if $x$ is a number then by definition $x^2=x\times x$; in other words, when we write "$x^2$", this is just a shorthand for $x\times x$. The phrase "identity" is used in different ways, but usually it refers to when a certain statement holds for all something; e.g. we sometimes say that $(x+1)^2\equiv x^2+2x+1$ is an identity because it holds for all numbers $x$, not just a particular vlaue of $x$. – Joe Sep 27 '24 at 17:46
  • @Joe Thank you very much for your comment. If I get it correctly, we could then write $x^2:=x\times x$. What I am still not seeing is what would prevent us from writing $(x+1)^2:=x^2+2x+1$. Any further help? – EoDmnFOr3q Sep 27 '24 at 17:49
  • 1
    This is not quite true. Of course $x^2:=x\cdot x$ is a definition of the exponent $2$. But then $(x+1)^2$ is not a new definition. It is already $(x+1)(x+1)$ by the previous definition (rewrite $y=x+1$ and apply the definition of $y^2:=y\cdot y$). So $(x+1)^n=x^n+\binom{n}{1}x^{n-1}+\cdots +1$ is the binomial identity, and not the binomial definition. – Dietrich Burde Sep 27 '24 at 17:52
  • @DietrichBurde Thank you very much for your comment. I think I might be able to see the difference, but it is so subtle… – EoDmnFOr3q Sep 27 '24 at 17:54
  • 1
    Yes to your second sentence. The reason it would be wrong to write $(x+1)^2:=x^2+2x+1$ is because even though this statement is true, it's not true by definition. The meaning of $(x+1)^2$ is $(x+1)(x+1)$, and to show that $(x+1)(x+1)=x^2+2x+1$ requires you to use facts about multiplication, like $a(b+c)\equiv ab+ac$ (also known as distributivity). – Joe Sep 27 '24 at 17:55
  • @Joe Thank you again for your comment. I think I get it now. So, we use $=$ to indicate that the LHS and the RHS are equal for some value(s), we use $:=$ to indicate that the LHS and the RHS are equal by definition, and we use $\equiv$ to indicate that the LHS and the RHS are equal because of other definitions and/or properties. Is this view more-or-less correct? – EoDmnFOr3q Sep 27 '24 at 17:57
  • 1
    I'm glad I could help. Yes, what you wrote is indeed more or less correct. A few notes: sometimes people use $=$ in the context of identities (e.g. they would write "$(x+1)^2=x^2+2x+1$", or, if they were being more careful, they would write "$(x+1)^2=x^2+2x+1$ for all values of $x$"); the notation := is inherently asymmetric – while it is true that $x^2:=x\times x$, it is not true that $x\times x:= x^2$ [...] – Joe Sep 27 '24 at 18:07
  • 1
    [...] I suppose you could write $x^2\equiv x\times x$, since it does hold for all values of $x$, but it would be misleading. The reason that $x^2$ always equals $x\times x$ is because of the meaning of the notation $x^2$, not because of any deep mathematical fact. You might want to look at the post Why do we not have to prove definitions? for some further discussion. – Joe Sep 27 '24 at 18:08
  • @Joe Thank you again! This further clarifies my doubts. – EoDmnFOr3q Sep 27 '24 at 18:11
  • 1
    Identity vs. conditional equation vs. inconsistent equation – ryang Nov 04 '24 at 12:40

0 Answers0