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5

You will know that these are mathematical statements when you can assign a truth value to them. That is, if you can look at it and say "that is true!" or "that is false!" then it is a mathematical statement. It doesn't mean anything else, it doesn't require numbers or symbols are anything commonly designated as "mathematical."


5

The solution is nothing more than some computation and observing $2\cdot27 = 90 - 36.$ \begin{align*} \tan^2(27^\circ) + 2 \tan(27 ^ \circ) \tan (36^\circ) &= \tan^2(27^\circ) + 2 \tan(27 ^ \circ) \cot (54^\circ) \\[1ex] &= \tan^2(27^\circ) + 2 \, \frac{\tan(27 ^ \circ)}{ \tan(54^\circ)} \\[1ex] &= \tan^2(27^\circ) + 2 \, \frac{\tan(27 ^ ...


5

Dan Willard published several papers about this topic in the Journal of Symbolic Logic. One place to start is the short Wikipedia article "Self-verifying theories". I am not familiar with the detailed proofs about Willard's theories, but when I have heard him talk about them he indicated they do not prove that multiplication is a total function, and in that ...


4

Make a truth table! We consider 4 cases. 1) $A$ true, $B$ true. Then $A \implies B$ is true, and ~$B \implies $~$A$ is true. 2) $A$ true, $B$ false. Then $A \implies B$ is false, and ~$B \implies $~$A$ is false. 3) $A$ false, $B$ true. Then $A \implies B$ is true, and ~$B \implies $~$A$ is true. 4) $A$ false, $B$ false. Then $A \implies B$ is true, ...


3

Robert's hint is superb. Full solution: If $k$ is the leftmost digit of $n$ then the number resulting from removing the leftmost digit of $n$ is congruent to $n-k\bmod 9$. Hence if we add $k$ to this number we obtain a number congruent to $n\bmod 9$. Conclusion: The congruence $\bmod 9$ is invariant under the operation of removing the leftmost digit of $n$ ...


3

In a realm of discourse (say, "ordinary mathematics", or ZFC) where both are provably true, or provably false, or each is provably true or false assuming that the other is true or false, they're equivalent. Tacitly assuming that we're in such a realm (since FLT has now been proven), you're correct. They're not semantically equivalent statements, though ...


2

Example: Consider the statement: If it is raining, then it is cloud. $Raining \implies Cloudy$ This does not mean that rain causes cloudiness, or that cloudiness causes rain. Neither is the case. It means only that it is not both raining and not cloudy. $\neg [Raining \land \neg Cloudy]$ In mathematics then, if not in everyday usage, $A\implies B\equiv ...


2

Your claim, "the mentioned statements are equivalent under only one circumstance that the only reason for boiling water is 100 degree Celsius temperature" is false. They are equivalent anyway. Suppose water boils when it is $100$ degrees OR [property $A$] is met. We claim "If the water is $100$ degrees, then it is boiling." and "If the water is not ...


2

A sentence is called mathematically acceptable statement if it is either true or false but not both. 1. All primes are odd numbers. 2. Two plus two is four. In the above sentences 1. is false and 2. is true and hence both of them are mathematical statements. 3. The sum of $x$ and $y$ is greater than 0. 4. Tomorrow is Friday. Above sentences 3. ...


2

A statement (or proposition) is a sentence that is either true or false. In your examples, which ones are true or false and which ones do not have such binary characteristics, i.e they cannot be described as being true or false? For example, me stating every integer is either even or odd is a statement that is either true or false. But $5+n$ is just an ...


2

By the Unique Factorization Theorem, it is enough to prove the result for integers $a, b, c$ which are each powers of a prime $p$. Let $a=p^\alpha$, $b=p^\beta$, and $c=p^\gamma$. Without loss of generality we may assume that $0\le \alpha\le \beta\le \gamma$. Then $\text{lcm}(ab,bc,ca)=p^{\beta+\gamma}$ and $\gcd(a,b,c)=p^\alpha$, so the product is ...


2

I don't think, that "rigorous" proof necessarily means that you provide a lot of details. It should be very clear, what you assume to be known, and what your definitions are, and how exactly the result is derived from this. For example, the first proof seems not to be very rigorous. First of all, we do not know how the torus is defined. If $T^n=S^1\times ...


1

A rigorous proof is a proof that can be seen to be valid by means of a valid proof-checking algorithm. Aristotle and many who followed showed us that certain forms of argument cannot lead from true premises to a false conclusion. One example of that is that a conclusion is validly deduced if it is true in every row in a truth table in which the premises ...


1

I think he means that if we chose all sequences in the cut rule $$\frac{\underline{A}\vdash C,B\quad\underline{A}',C\vdash\underline{B'}}{\underline{A},\underline{A}'\vdash\underline{B},\underline{B}'}$$ to be empty, then we can informally read it as '($\vdash C$ a.k.a. $C$-on-the-right) and ($C\vdash$ a.k.a. not $C$-on-the-left) implies ($\vdash$ a.k.a. ...


1

using a web search, this one looks interesting and maybe meets your needs: http://us.metamath.org/mpegif/mmset.html


1

If you're interpreting "if...then..." intuitively when thinking about logic you're going to confuse yourself. $p \implies q$ (i.e. "$p$ implies $q$") has a very particular meaning in propositional logic, which is essentially that the same as $\neg(\neg p \land q)$ (i.e. "never not p and q"). To express this fact the type of implication we use in ...


1

In Logic, "If A, then B" is often written as $$A\implies B,$$ which can be read as"A implies B". This relation $\implies$ is called "material implication" and is defined as a function of the truth values of A and B: Whenever A is false, $A\implies B$ is defined to be true. Whenever B is true, $A\implies B$ is defined to be true. The only remaining ...



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