# Difference of ≡ and ⇔ [duplicate]

I'm studying cs in germany, I have been going through my script starting with propositional logic. I don't really understand the difference between those. I watched some YT-Videos on the topic but i didn't really helped me differentiate between those. They just sometimes use one or the other.

"Definition" in my own words (my understanding):

• The script says something like "$$A ⇔ B$$" gets when "$$A ↔ B$$" is true. So only if both statements A, B are true or both are false "$$A ⇔ B$$" would get used.
• The Term "$$A ≡ B$$" gets used if two logical Terms (consisting of Variables, Constants, ...) result in a equal logical value.

Example:
$$A ↔ B ⇔ (A → B) ∧ (B → A)$$
$$A ∧ (B ∧ C) ≡ (A ∧ B) ∧ C$$.

Problem:When given to "equal" statements they will be right and wrong the same time therefore i can use $$⇔$$. But they also will have the same logical value so i can use $$≡$$. I can't really differentiate between them.
Almost a different question. Why do the de morgan laws use ↔, why don't the use one of $$≡, ⇔$$? $$¬(A ∧ B) ↔ (¬A ∨ ¬B)$$. English is not my first language and i translated my question mostly by myself. Also its my first time asking a question here.

• I've always used them as basically different symbols for the same concept. There might be some reasons to distinguish them, but I have never needed them to be distinguished. I guess you could define $a\iff b$ as $(a\to b)\land (b\to a),$ while we could define $a\equiv b$ as $(a\land b)\lor(\lnot a\land\lnot b).$ Maybe it requires excluded middle to prove these are equivalent? Commented Jun 13 at 20:48
• Observe that your bullet point summary of ⇔ ↔ ≡ actually says that they are thoroughly interchangeable. They do have different meanings, but even when this is respected, frequently the choice between them is a matter of emphasis anyway. See whether the links in this answer's addendum help make things a bit clearer. Commented Jun 14 at 5:23

It will be question of convention on notation, if you want to distinguish two operators, one can set the following.

Indeed here we can note two things :

Preliminary before math

Take two sentences :

• $$X=$$ "The first day of the year is 1st January"
• $$Y=$$ "The Earth is a planet"

Both $$X$$ and $$Y$$ are sentences. They have both a meaning different and independant of each other. And they are both true.

You see here, that we cannot have $$X \implies Y$$ or $$Y \implies X$$, it doesn't make any sense. It has no meaning. Recall that we use the notation $$X ⇔ Y$$ to say ($$X \implies Y$$ and $$Y \implies X$$).

So because there are not logical implication (very concrete in very real life), between $$X$$ and $$Y$$. So they can't be equivalent (so "$$X ⇔ Y$$" is not true). Do you agree with that concrete example ?

Yet both $$X$$ and $$Y$$ are both true,the first day of the year is 1st January and the Earth is a planet :

So $$X \equiv Y$$

Meaning is always about a system, a language, which has a real, a concret meaning. Sometimes this language is English, French, German, sometimes its mathematic. Yet, in every language something can be true or can be false (or neither true nor false).

First

$$⇔ \text{and} \leftrightarrow$$

are the same object just noted differently.

When you note $$A ⇔ B$$ it means they have not only the same logical value but also the same meaning.

For example : $$x\in[a,b] ⇔ a \leq x \leq b$$

Secondly

The sign $$\equiv$$ is just here to say that thing have the same logical value (in binary logic, True or False).

For example :

$$(\forall \in \mathbb{N}^*, n>1, n-1>0) \equiv (\pi=\pi)$$

Both are true, but have of course not the same meaning.

Moreover

You see that :

$$A= \ "[(\forall \in \mathbb{N}^*, n>1, n-1>0) ⇔ (\pi=\pi)]"$$

is false.

Yet :

$$B= \ "[(\forall \in \mathbb{N}^*, n>1, n-1>0) \equiv (\pi=\pi)]"$$

is true.

• Can you find a case where the distinction is a real difference? Commented Jun 13 at 20:50
• Between what and what ?
– EDX
Commented Jun 13 at 20:50
• A case where only one of $P\iff Q$ and $P\equiv Q$ is true? Commented Jun 13 at 20:52
• Ok get your question, yes.
– EDX
Commented Jun 13 at 20:53
• you loose me at your second example. Whats the difference between meaning and logical value? Im not saying its wrong... Commented Jun 13 at 21:03