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arthurhenry
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Is it true that every matrix is similar to its transpose? A claim in Wikipedia...
(field is alg. closed)
(field is alg. closed)
Yes, in the context of mathematical logic, a statement can be both true and false at the same time. This is known as a contradiction and is typically denoted as "A and not A" or "A ~Trans(A)". However, in everyday language and reasoning, a statement cannot be both true and false simultaneously.
"A ~Trans(A)" and "A implies ~A" are two different statements with different meanings. "A ~Trans(A)" means that A is not equivalent to its negation, while "A implies ~A" means that if A is true, then its negation must also be true. The former is a statement about the logical relationship between A and its negation, while the latter is a conditional statement.
Yes, the truth value of a statement can vary depending on the context in which it is evaluated. For example, the statement "It is raining" can be true in one location but false in another location. Similarly, the statement "A ~Trans(A)" can be true in one logical system but false in another.
"A ~Trans(A)" is essentially the negation of the law of non-contradiction, which states that a statement cannot be both true and false at the same time. In other words, "A ~Trans(A)" is equivalent to "not(A and not A)", which is the logical form of the law of non-contradiction.
"A ~Trans(A)" is a statement that falls under the realm of mathematical logic and cannot be proven or disproven in the same way that empirical claims can be. Instead, it can only be evaluated based on the logical rules and axioms of the system in which it is being considered. In some logical systems, "A ~Trans(A)" may be considered a valid statement, while in others it may be considered contradictory.