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 Apr24 comment What is the answer to this hard problem for 4th graders? What kind of 4th grade teacher would ask this question? Mar26 comment Are variables logical or non-logical symbols in a logic system? Perhaps this will help. Mar26 comment What does $\in$ mean? Perhaps this will help. Mar25 comment Could somebody please help me prove this using the properties of real numbers introduced in elementary algebra? I think there is a typo in your sentence "Your question is (basically): if a $\in$ F, how do we know that a/1 = 1?" I believe the question asks how do we know that a/1 = a. Feb13 revised When an equation has no solutions, denote it with $x\in\varnothing$. added 442 characters in body Feb13 comment How do you read the symbol “$\in$”? Simply read it as "what y can be replaced by belongs to" Feb9 comment What to answer when people ask what I do in mathematics Trolls don't last very long in the real world. Feb9 comment What to answer when people ask what I do in mathematics Because you are not being true to thyself. Jan23 revised When an equation has no solutions, denote it with $x\in\varnothing$. added 388 characters in body Jan22 awarded Peer Pressure Jan22 comment When an equation has no solutions, denote it with $x\in\varnothing$. chat.stackexchange.com/rooms/20486/… Jan22 comment When an equation has no solutions, denote it with $x\in\varnothing$. Do you agree it is an "unknown"? Jan22 comment When an equation has no solutions, denote it with $x\in\varnothing$. When they say that "a variable is a mathematical object" they mean that it is an object used in mathematics to represent one or more numbers. If no numbers exist for it to represent, then the set it represents is empty. The variable does not exist in the set. The numbers it represents exist in the set. A variable is a symbol used to represent one or more numbers. Those numbers are in a set. Not the variable itself, it is just a letter. Jan22 comment When an equation has no solutions, denote it with $x\in\varnothing$. First of all $x$ is not automatically something - a variable denotes an "unknown thing" we don't know whether anything exists in the solution set. The solution set here does not contain a variable. The solution set contains values of the variable that make the equation a true statement. Jan22 comment When an equation has no solutions, denote it with $x\in\varnothing$. Okay, let's use $0\cdot x = 0$ instead :-) Jan22 comment When an equation has no solutions, denote it with $x\in\varnothing$. recall: unknown quantities can be represented by words such as 'thing', etc. Also, drawings of squares etc. can be used to symbolize unknowns, and finally a letter, such as "x", can represent an unknown. In this case the "unknown quantities" has no value, therefore our our symbol that represents it can have no value. The other extreme is found in the case of the solution set of $x = 0/0$ over the set of real numbers; the solution set for x is the set of all real numbers that is to say x belongs to the set of all real numbers. Jan22 comment When an equation has no solutions, denote it with $x\in\varnothing$. Okay, $x$ is 'something', or simply a 'thing'. Now, what that 'thing' is depends completely on what value make the equation a true statement. Since no value makes the equation a true statement the "thing" can have no value. The same argument is used when considering the solution set of $x=1/0$ over the set of real numbers; the solution set for x is the empty set that is to say x belongs to the empty set. Jan8 awarded Quorum Jan7 awarded Enthusiast Jan5 comment How to stop forgetting proofs - for a first course in Real Analysis? The best way to stop forgetting is to only remember the analogies between the proofs.