query: Let $R,S \subseteq A \times A$ be transitive binary relations. If $R\circ S=S\circ R$ then $R\circ S$ is also transitive.
positive_passages.docid: 25155
positive_passages.text: Suppose $R$ and $S$ are transitive relations on $A$. Prove that if $S \circ R \subseteq R \circ S$ then $R \circ S$ is transitive.
negative_passages.docid: 71977
negative_passages.text: Give an example of a relation R on $A^2$ which is reflexive, symmetric, and not transitive
negative_passages.docid: 41099
negative_passages.text: If $A \subseteq C$ and $B \subseteq D$ then $A \times B \subseteq C \times D$
negative_passages.docid: 74015
negative_passages.text: Number of relations that are both symmetric and reflexive
negative_passages.docid: 116866
negative_passages.text: $A\times B$ connected component implies $A,B$ connect components
negative_passages.docid: 53101
negative_passages.text: Is it true that the relation |A| < |B| is a sufficient condition for claiming that $f$ is a bijection?
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: 57418
negative_passages.text: How to prove this statement: $\binom{r}{r}+\binom{r+1}{r}+\cdots+\binom{n}{r}=\binom{n+1}{r+1}$
negative_passages.docid: 63267
negative_passages.text: Under what conditions on the sets $A$ and $B$ will it be true that $A\times B= B\times A$
negative_passages.docid: 13906
negative_passages.text: How to prove the set $S$ is closed under the binary operation $a*b=a+b+kab$?
negative_passages.docid: 146231
negative_passages.text: Let X be a set with an associative binary operation ◦ with identity e. Suppose every element of X satisﬁes x ◦ x = e. Prove that ◦ is commutative.
negative_passages.docid: 56399
negative_passages.text: Proof for $A \cup B = B$ if and only if $A \subset B$
negative_passages.docid: 109070
negative_passages.text: Question about the proof of the distributive property for sets
negative_passages.docid: 70278
negative_passages.text: $A\subseteq B\to C\setminus B\subseteq C\setminus A\,$ -- how to prove this?
negative_passages.docid: 51206
negative_passages.text: If $X$ and $Y$ are sets, their cartesian product $X\times Y$ is a set.
negative_passages.docid: 52949
negative_passages.text: If $aRb \land bRc$ never occurs, is a relation considered transitive?
negative_passages.docid: 13026
negative_passages.text: Prove that if AB=I_m, then rk(A)=rk(B).