- #1
zeromodz
- 244
- 0
If two are equal to a third, do they all equal each other? Can you prove this?
Proving Equality in Math: Solving the Question of Two Equaling a Third
- Context: High School
- Thread starter zeromodz
- Start date
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Discussion Overview
The discussion revolves around the question of whether two entities that are equal to a third entity must also be equal to each other. Participants explore the implications of this question within the context of mathematical equality and the transitive property.
Discussion Character
- Debate/contested
- Conceptual clarification
- Mathematical reasoning
Main Points Raised
- Some participants suggest that the question can be addressed using the transitive property of equality.
- One participant interprets the question as asking if two numbers equal to 1/3 must themselves be equal, arguing that they could be different values that sum to 1/3.
- Another participant agrees with this interpretation, noting that the ambiguity of the question affects its clarity.
- A participant introduces the principle of explosion, explaining that if a false statement is accepted, it can lead to any statement being proven true, illustrating this with an algebraic example.
- One participant critiques the original phrasing of the question, emphasizing the importance of defining the objects involved and the nature of equality being discussed.
Areas of Agreement / Disagreement
Participants express differing interpretations of the initial question, with no consensus on the precise meaning or implications of equality in this context. Some agree on the need for clarity in definitions, while others present competing views on the nature of the relationships described.
Contextual Notes
The discussion highlights the ambiguity in the phrasing of the question and the necessity of defining terms such as "equal" and the specific entities involved. Participants also note the potential for different mathematical frameworks to influence interpretations.
- #2
nicksauce
Science Advisor
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Depending on what you mean by "two" and "a third", this fact follows from the transitive property.
- #3
Mentallic
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I'm going to assume it means if two numbers are equal to 1/3, are the numbers equal?
Well, not necessarily. For them to be equal that means each number has to be 1/6, but of course you can give each number any other value which still satisfies that they add to 1/3.
Well, not necessarily. For them to be equal that means each number has to be 1/6, but of course you can give each number any other value which still satisfies that they add to 1/3.
- #4
Mark44
Mentor
- 38,109
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The question seems to be ambiguous. I interpret it as nicksauce does, to mean "if two numbers are equal to a third number, then all three are equal."
- #5
- 12,179
- 186
You're interpreting this to mean the sum of two numbers has a value equal to 1/3. Mark44 and nicksauce interpret it to mean that two numbers are each, themself, equal to a third number.Mentallic said:
- #6
Mentallic
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Oh yeah that seems like a much more reasonable interpretation 
- #7
- 12,179
- 186
It's better than my initial thought of "2 = 1/3" 
- #8
disregardthat
Science Advisor
- 1,864
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It's called the principle of explosion. If you grant a false statement, then you can logically prove that any statement is true. Hence your system "explodes" so that the truth values of all statements are 1 (and 0). So if 2 = 1/3, then a = b for any numbers a and b. This can however be proved by mere algebraic manipulation. From 2 = 1/3 you will get 6 = 1, so 5 = 0, or 1 = 0, and from there a - b = 0, so a = b.
EDIT: Oh, after reading the other comments, I realize that the question really was whether a = c, b = c implies a = b. That has been explained.
EDIT: Oh, after reading the other comments, I realize that the question really was whether a = c, b = c implies a = b. That has been explained.
Last edited:
- #9
slider142
- 1,016
- 72
Your original sentence fragment is missing its subject. I assume you mean "If two [objects] are equal to a third [object], ..."
This is true as long as your definition of "equal" shares the transitive property, which is included in the usual mathematical definition of an equivalence between objects.
Specifically, the transitive porperty states that if the relationship between three objects, {a, b, c}, is denoted with the sign "=" such that a = c and b = c, then it follows that a = b.
However, I do not know what your definition of equality is, or what 3 objects you are referring to.
It is quite possible that you are talking about a different kind of equality, which is mathematically more like placing objects into sets.
That is, "A chicken is an animal.", and "A dog is an animal." do not enjoy the transitive property that "A chicken is a dog."
This is true as long as your definition of "equal" shares the transitive property, which is included in the usual mathematical definition of an equivalence between objects.
Specifically, the transitive porperty states that if the relationship between three objects, {a, b, c}, is denoted with the sign "=" such that a = c and b = c, then it follows that a = b.
However, I do not know what your definition of equality is, or what 3 objects you are referring to.
It is quite possible that you are talking about a different kind of equality, which is mathematically more like placing objects into sets.
That is, "A chicken is an animal.", and "A dog is an animal." do not enjoy the transitive property that "A chicken is a dog."
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