#### Please [explain like I’m 5 years old](https://www.howtogeek.com/694298/what-does-eli5-mean-and-how-do-you-use-it), compare and contrast ALL 4 symbols =, ≡, ↔, ⇔.
All answers beneath are too knotty and byzantine to my secondary school daughter.
– https://math.stackexchange.com/q/500644
– https://math.stackexchange.com/q/3926367
– https://math.stackexchange.com/q/2432462
## https://math.stackexchange.com/q/4645583
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An equal sign is used to show a value is equal to another. 1+1=2
The equivalence sign is used to define one thing as equal to another. e is defined as (tribar symbol) lim x->infinity (1+1/x)^x
The material biconditional is the same as the equivalence sign.
The logical biconditional is just “if and only if” whatever is on one side must have the same truth value as the other side. It’s essentially an equal sign for boolean values.
An equal sign is used to show a value is equal to another. 1+1=2
The equivalence sign is used to define one thing as equal to another. e is defined as (tribar symbol) lim x->infinity (1+1/x)^x
The material biconditional is the same as the equivalence sign.
The logical biconditional is just “if and only if” whatever is on one side must have the same truth value as the other side. It’s essentially an equal sign for boolean values.
These are all symbols that assert a *relation* between two objects. Relationships can include things like equality (`=`), greater than (`>`), taller than, heavier than, has more calories than, etc.
`=` most frequently is used to indicate *equality.* What equality means depends on the context, but for in the context of numbers, and the sake of simplicity you can take it to mean “numerically equal” (the same number). That’s not really what it means, but in order to define the formal definition of equality we’d have to start talking about a particular axiom system, and that’s not ELI5.
`≡` often means equivalence or congruence, which has different meanings in different contexts. In geometry, you might say two different triangles are congruent to each other—what that means has its own definition. In modular arithmetic, you might say two expressions are equivalent “modulo some integer.” Again, the definition of equivalence varies depending on how you’re using it.
`↔` Often means “if and only if,” also known as a biconditional. It is used in logical propositions. I.e., it is part of the vocabulary in the language of making claims or statements that are either true or false. An example is “Bob will eat the ice cream if and only if it is vanilla flavored.”
In order to understand biconditionals, it’s helpful first to understand the conditional `→`, which indicates an “if…then” relationship. For example, “If Sparky is a dog, then Sparking has four legs.”
`⇔` Is very similar, but has to do with logical implication. Implication means one statement follows from another. A two way implication would mean a statement follows from another, and vice versa. The difference is `→` relates two expressions, and `⇒` relates two statements.
I’ll just add more examples for some parts since other people explained it pretty well.
The triple bar for equivalence is especially useful for modular arithmetic. An example of that is on a clock, 4 am and 4 pm are equivalent so comparing 24 hour time to 12 hour time, 4 is equivalent to 16 (mod 12).
A logical biconditional means “if and only if” so the two things necessarily mean the other is true. A shape is a triangle if and only if it has exactly 3 sides. That means that any shape with exactly 3 sides is a triangle and a triangle has exactly 3 sides.
These are all symbols that assert a *relation* between two objects. Relationships can include things like equality (`=`), greater than (`>`), taller than, heavier than, has more calories than, etc.
`=` most frequently is used to indicate *equality.* What equality means depends on the context, but for in the context of numbers, and the sake of simplicity you can take it to mean “numerically equal” (the same number). That’s not really what it means, but in order to define the formal definition of equality we’d have to start talking about a particular axiom system, and that’s not ELI5.
`≡` often means equivalence or congruence, which has different meanings in different contexts. In geometry, you might say two different triangles are congruent to each other—what that means has its own definition. In modular arithmetic, you might say two expressions are equivalent “modulo some integer.” Again, the definition of equivalence varies depending on how you’re using it.
`↔` Often means “if and only if,” also known as a biconditional. It is used in logical propositions. I.e., it is part of the vocabulary in the language of making claims or statements that are either true or false. An example is “Bob will eat the ice cream if and only if it is vanilla flavored.”
In order to understand biconditionals, it’s helpful first to understand the conditional `→`, which indicates an “if…then” relationship. For example, “If Sparky is a dog, then Sparking has four legs.”
`⇔` Is very similar, but has to do with logical implication. Implication means one statement follows from another. A two way implication would mean a statement follows from another, and vice versa. The difference is `→` relates two expressions, and `⇒` relates two statements.
I’ll just add more examples for some parts since other people explained it pretty well.
The triple bar for equivalence is especially useful for modular arithmetic. An example of that is on a clock, 4 am and 4 pm are equivalent so comparing 24 hour time to 12 hour time, 4 is equivalent to 16 (mod 12).
A logical biconditional means “if and only if” so the two things necessarily mean the other is true. A shape is a triangle if and only if it has exactly 3 sides. That means that any shape with exactly 3 sides is a triangle and a triangle has exactly 3 sides.
= is used to signify two numerical quantities are equal. 5=6-1.
≡ is used in lieu of “is the same as”, verbally speaking. It denotes identities, i.e. two expressions whose value is equal no matter what the variables are. For example: (a+b)^2 ≡ a^2 + 2ab + b^2 . In most cases, we simply use = instead, but it is important to remember that it’s simply an equivalence sign.
Do note that ≡ is more frequently used to denote **congruence**. Two numbers are congruent modulo a number N if N divides their difference. We write a ≡ b (mod n).
⇔ is “if and only if”. P ⇔ Q means Q is true if and only if P is true. The truth table for this condition is such that P ⇔ Q is true in two cases: both P and Q are true, or both P and Q are false. This logical equivalence **can also be written**, albeit less frequently, as P ≡ Q
↔ is used exactly the same way as ⇔. No difference whatsoever.
Summary: It’s better to learn these symbols as part of the required reading, there sometimes is significant overlap between what they are used for.
I think that maybe these explanations might be a bit hard to understand, and so I would like to make them simpler. Also, due to the close relationship between these four concepts, many people will tell you that some are the same as others when, in fact, they are not *quite* the same. It is not uncommon for this to be done in the classroom or in books on math and logic to simplify things, and it is quite possible that the distinction can be lost on, say, a teacher teaching the concepts. However, as these are stated as four distinct things I think that the nitpicky details are being asked for.
(1)= Equal Sign
This is equal to that. Example: 2+2=4
(2) ≡ Equivalence, Identity
If A ≡ B, then if A is true B is true and if A is false B is false. This is often described as “if and only if” and is sometimes written as iff. Example: You get $100 at the end of the semester ≡ Your GPA is over 3.5.
The next two can be a bit tricky as they are not always used exactly the same way in all math and logic. Additionally, both are forms of equivalence, and are thus often used interchangeably with equivalence and each other.
(3) ↔ Material BiConditional
This is used when two statements are connected together in equivalence, such that one is true if and only if the other is true at this time, but not necessarily at a different time or under different circumstances.
Example: Due to a snowstorm this morning (the material condition), Johnny cannot get to school from his house today unless his Dad drives him. Further, once there he will not be able to leave until the end of the school day.
Thus, “Johnny is at school” ↔ “Johnny was driven to school by his father this morning”.
On any other day, this may or may not be true, and thus this is material rather than logical.
(4) ⇔ Logical BiConditional
This is when two statements are connected together in equivalence such that one is true if and only if the other is true, period, regardless of any material conditions.
Example: A is equal to 4 ⇔ A is equal to 2+2. If the first part is true, then the second part is, and if the second part is not, neither is the first. No material changes can make that not the case, so it is a logical bi-conditional and not merely a material bi-conditional.
I tried to make this as clear and precise as possible. I hope it helps.
= is used to signify two numerical quantities are equal. 5=6-1.
≡ is used in lieu of “is the same as”, verbally speaking. It denotes identities, i.e. two expressions whose value is equal no matter what the variables are. For example: (a+b)^2 ≡ a^2 + 2ab + b^2 . In most cases, we simply use = instead, but it is important to remember that it’s simply an equivalence sign.
Do note that ≡ is more frequently used to denote **congruence**. Two numbers are congruent modulo a number N if N divides their difference. We write a ≡ b (mod n).
⇔ is “if and only if”. P ⇔ Q means Q is true if and only if P is true. The truth table for this condition is such that P ⇔ Q is true in two cases: both P and Q are true, or both P and Q are false. This logical equivalence **can also be written**, albeit less frequently, as P ≡ Q
↔ is used exactly the same way as ⇔. No difference whatsoever.
Summary: It’s better to learn these symbols as part of the required reading, there sometimes is significant overlap between what they are used for.
I think that maybe these explanations might be a bit hard to understand, and so I would like to make them simpler. Also, due to the close relationship between these four concepts, many people will tell you that some are the same as others when, in fact, they are not *quite* the same. It is not uncommon for this to be done in the classroom or in books on math and logic to simplify things, and it is quite possible that the distinction can be lost on, say, a teacher teaching the concepts. However, as these are stated as four distinct things I think that the nitpicky details are being asked for.
(1)= Equal Sign
This is equal to that. Example: 2+2=4
(2) ≡ Equivalence, Identity
If A ≡ B, then if A is true B is true and if A is false B is false. This is often described as “if and only if” and is sometimes written as iff. Example: You get $100 at the end of the semester ≡ Your GPA is over 3.5.
The next two can be a bit tricky as they are not always used exactly the same way in all math and logic. Additionally, both are forms of equivalence, and are thus often used interchangeably with equivalence and each other.
(3) ↔ Material BiConditional
This is used when two statements are connected together in equivalence, such that one is true if and only if the other is true at this time, but not necessarily at a different time or under different circumstances.
Example: Due to a snowstorm this morning (the material condition), Johnny cannot get to school from his house today unless his Dad drives him. Further, once there he will not be able to leave until the end of the school day.
Thus, “Johnny is at school” ↔ “Johnny was driven to school by his father this morning”.
On any other day, this may or may not be true, and thus this is material rather than logical.
(4) ⇔ Logical BiConditional
This is when two statements are connected together in equivalence such that one is true if and only if the other is true, period, regardless of any material conditions.
Example: A is equal to 4 ⇔ A is equal to 2+2. If the first part is true, then the second part is, and if the second part is not, neither is the first. No material changes can make that not the case, so it is a logical bi-conditional and not merely a material bi-conditional.
I tried to make this as clear and precise as possible. I hope it helps.
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