Transitive relation

Step 1: First ((p,q)) and ((q,r)) imply ((p,r)). Step 2: Then ((p,r)) and ((r,s)) imply ((p,s)), and ((p,s)) with ((s,t)) implies ((p,t)). Step 3: Apply transitivity repeatedly in a long chain.

Step 1: In a difference of relations, some required pairs may be removed. Step 2: After removal, two linked remaining pairs may miss their required third pair. Step 3: Transitivity is not automatically preserved under difference.

Step 1: (R\cup S) contains both ((2,3)) and ((3,4)). Step 2: Transitivity requires ((2,4)), but it is not in the union. Step 3: Chains from two different relations can create a new missing pair in the union.

Step 1: ((a,c)\in R\circ R) when some (b) exists with ((a,b)\in R) and ((b,c)\in R). Step 2: Since (R) is transitive, such ((a,c)) belongs to (R). Step 3: For composition questions, read the definition through ordered pairs.

Step 1: (R\circ R\subseteq R) means the pair formed by two linked pairs is again in (R). Step 2: This is exactly the condition of transitivity. Step 3: Learn to connect composition language with the definition of transitivity.

Step 1: ((1,2)) and ((2,3)) give the required ((1,3)). Step 2: With ((3,5)), ((1,3)) requires ((1,5)), and ((2,3)) requires ((2,5)); both are present. Step 3: Match every required direct jump in long chains.

Step 1: ((1,5)) and ((5,3)) require ((1,3)). Step 2: This pair is not in the list, so it must be added. Step 3: Include existing pairs like ((1,5)) in further checking.

Step 1: Try to find a chain where the common factor changes. Step 2: (2) and (6) have common divisor (2), and (6) and (9) have common divisor (3), but (2) and (9) have no common divisor greater than (1). Step 3: In hard number relations, choose counterexamples carefully.

Step 1: (2) and (3) are coprime, and (3) and (4) are also coprime. Step 2: But (2) and (4) are not coprime. Hence transitivity fails. Step 3: In coprime relations, the middle number may connect for different reasons.

Step 1: If (a-b) is an integer and (b-c) is an integer, their sum (a-c) is also an integer. Step 2: Hence (aRc) is true. Step 3: In difference relations, remember closure under addition.

Step 1: If (a-b>2) and (b-c>2), adding gives (a-c>4). Step 2: Since (a-c>4) implies (a-c>2), transitivity holds. Step 3: Adding inequalities can give a stronger conclusion.

Step 1: Take (a=3), (b=1.5), and (c=0). Step 2: (0<3-1.5<2) and (0<1.5-0<2), but (3-0=3), which is not less than (2). Step 3: In bounded inequalities, two small gaps can cross the bound.

Step 1: ((1,2)) and ((2,3)) imply ((1,3)). Step 2: Then ((1,3)) and ((3,1)) imply ((1,1)). Step 3: Cyclic chains produce self-pairs.

Step 1: The three pairs form a cycle. Step 2: Repeated transitivity gives ((1,3),(2,1),(3,2)) and then self-pairs like ((1,1),(2,2),(3,3)). Step 3: The transitive closure of a full cycle often connects the whole small set.

Step 1: ((1,3)) and ((3,1)) require ((1,1)), which is present. Step 2: ((3,1)) and ((1,3)) require ((3,3)), also present. Step 3: Reverse pairs do not break transitivity if both required self-pairs are present.

Step 1: ((1,3)) and ((3,1)) require ((1,1)), and ((3,1)) with ((1,3)) requires ((3,3)). Step 2: Both are present; ((2,4)) does not form any further linked chain. Step 3: An isolated pair does not break transitivity by itself.

Step 1: ((3,1)) and ((1,3)) require ((3,3)). Step 2: ((3,3)) is not in the relation, so transitivity fails. Step 3: Reverse pairs must be checked in both orders.

Step 1: ((a,b)) and ((b,c)) imply ((a,c)). Step 2: Then ((a,c)) and ((c,a)) imply ((a,a)); rotating the cycle gives ((b,b)) and ((c,c)) too. Step 3: Cyclic relations force self-pairs under transitivity.

Step 1: From (a\le 2b) and (b\le 2c), we only get (a\le 4c). Step 2: Transitivity requires (a\le 2c), which need not hold. Take (a=4), (b=2), (c=1); (4\le4), (2\le2), but (4\le2) is false. Step 3: In multiplier-based relations, the bound may change.

Step 1: If (a=kb) and (b=kc), then (a=k^2c). Step 2: Transitivity requires (a=kc), so generally (k^2=k); for positive (k), this gives (k=1). Step 3: In fixed-multiplier relations, apply the multiplier twice.