query: If $a, b$ are transcendental then $a+b$ is transcendental or $ab$ is transcendental
positive_passages.docid: 18465
positive_passages.text: Sum and product of two transcendental numbers can't be both algebraic
negative_passages.docid: 154822
negative_passages.text: Properties of the composition law $a @ b= ab + b + a$ on the real numbers
negative_passages.docid: 140167
negative_passages.text: Proving $A+A'B=A+B$ without truth tables
negative_passages.docid: 18693
negative_passages.text: If $a,b$ are positive integers such that $a^{n}+n$ divides $b^{n}+n$ for every $n$, then $a=b$
negative_passages.docid: 106442
negative_passages.text: Show that AB is singular if A is singular
negative_passages.docid: 118981
negative_passages.text: Proving $\sqrt{ab} = \sqrt a\sqrt b$
negative_passages.docid: 91285
negative_passages.text: Prove $a+b+c+d $ is composite
negative_passages.docid: 10348
negative_passages.text: Prove that if the $\gcd(a,b)=1$, then $\gcd(a+b,ab)=1$
negative_passages.docid: 36602
negative_passages.text: $a^m=b^m$ and $a^n=b^n$ imply $a=b$
negative_passages.docid: 22856
negative_passages.text: Ways to go from A to B(proof)
negative_passages.docid: 63535
negative_passages.text: Prove that if $A – C = B – C$ and $A\cap C = B\cap C$ then $A = B$
negative_passages.docid: 8787
negative_passages.text: In a ring $(A,+, \cdot)$ if $aba = a$ then $bab = b$ and all non zero elements in $A$ are invertible.
negative_passages.docid: 124320
negative_passages.text: $a,b,c>0 \text{ s.t. }a+b+c=1 \implies \sqrt{ab+c}+\sqrt{bc+a}+\sqrt{ca+b} \ge 1+ \sqrt{ab}+\sqrt{bc}+\sqrt{ca}$
negative_passages.docid: 33448
negative_passages.text: Proof if $AB+BA=0$ Then atleast one of the matrices are singular.
negative_passages.docid: 33542
negative_passages.text: When is $A- B = B- A$?
negative_passages.docid: 12096
negative_passages.text: If $a+1/a$ is an integer, then so is $a^t+1/a^t$ for $t\in\mathbb N$
negative_passages.docid: 15626
negative_passages.text: Relations of a~b iff b = ak^2