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The phrase denote «object» by «symbol» means the same thing as We will refer to «object» using the name «symbol». In English, it is permitted to “reverse” this to denote by «symbol» «object» without changing the meaning. I have also heard the usage denote «object» as «symbol», with identical meaning, but this is unusual, and I believe that it is technically incorrect as well. In any case, to omit by suggests that «symbol» is actually «object», and would be incorrect for this context.

ref: Merriam-Webster’s Learner’s Dictionary, denote
ref: Merriam-Webster dictionary on-line, denote
ref: Oxford dictionary on-line, denote
ref: Cambridge English Dictionary on-line, denote

The phrase denote «object» by «symbol» means the same thing as We will refer to «object» using the name «symbol». In English, it is permitted to “reverse” this to denote by «symbol» «object» without changing the meaning. I have also heard the usage denote «object» as «symbol», with identical meaning, but this is unusual, and I believe that it is technically incorrect as well. In any case, to omit by suggests that «symbol» is actually «object».

ref: Merriam-Webster’s Learner’s Dictionary, denote
ref: Merriam-Webster dictionary on-line, denote
ref: Oxford dictionary on-line, denote
ref: Cambridge English Dictionary on-line, denote

The phrase denote «object» by «symbol» means the same thing as We will refer to «object» using the name «symbol». In English, it is permitted to “reverse” this to denote by «symbol» «object» without changing the meaning. I have also heard the usage denote «object» as «symbol», with identical meaning, but this is unusual, and I believe that it is technically incorrect as well. In any case, to omit by suggests that «symbol» is actually «object», and would be incorrect for this context.

ref: Merriam-Webster’s Learner’s Dictionary, denote
ref: Merriam-Webster dictionary on-line, denote
ref: Oxford dictionary on-line, denote
ref: Cambridge English Dictionary on-line, denote

1
source | link

The phrase denote «object» by «symbol» means the same thing as We will refer to «object» using the name «symbol». In English, it is permitted to “reverse” this to denote by «symbol» «object» without changing the meaning. I have also heard the usage denote «object» as «symbol», with identical meaning, but this is unusual, and I believe that it is technically incorrect as well. In any case, to omit by suggests that «symbol» is actually «object».

ref: Merriam-Webster’s Learner’s Dictionary, denote
ref: Merriam-Webster dictionary on-line, denote
ref: Oxford dictionary on-line, denote
ref: Cambridge English Dictionary on-line, denote