When to use equal to or equivalent to?

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Discussion Overview

The discussion revolves around the appropriate usage of the symbols \equiv and = in mathematical contexts, exploring their meanings and implications in different scenarios. The scope includes conceptual clarification and technical explanation regarding mathematical notation.

Discussion Character

  • Conceptual clarification, Technical explanation

Main Points Raised

  • Some participants suggest that \equiv should be used for equations that are identically true, while = is reserved for equations that are conditionally true.
  • One participant provides examples, stating that y = ax + b represents an assignment, while a + b == b + a is a statement of equivalence.
  • Another participant emphasizes that equations like (x + 1)² \equiv x² + 2x + 1 and sin²(x) + cos²(x) \equiv 1 are always true for any real x, thus justifying the use of \equiv.
  • Conversely, they argue that an equation such as 2x + 1 = 5 is only true under specific conditions (when x = 2), which supports the use of =.

Areas of Agreement / Disagreement

Participants present differing views on the usage of \equiv and =, with some agreeing on the general principles while others provide examples that may blur the lines between the two symbols. The discussion does not reach a consensus on a definitive rule.

Contextual Notes

There are nuances in the definitions of "identically true" and "conditionally true" that are not fully explored, and the examples provided may depend on specific contexts or interpretations.

Darth Frodo
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When is it appropriate to use \equiv as opposed to =?
 
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well, here are a couple of examples, although I don't know that it would ever be wrong to use = when you mean equivalent.

y = ax + b [this is an "assignment" which is typical for "="]

a + b == b + a [this is not an assignment but a statement of equivalence]
 
Darth Frodo said:
When is it appropriate to use \equiv as opposed to =?
Use ##\equiv## for equations that are identically true, such as (x + 1)2 ##\equiv## x2 + 2x + 1 and sin2(x) + cos2(x) ##\equiv## 1. Each of these equations is true for any real x.

Use = for equations that are conditionally true, such as 2x + 1 = 5. This equation is true only for x = 2; i.e., only under certain conditions.
 
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Oh ok. Thanks Mark & phinds! That really cleared it up!
 

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