- #1
Terrell
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Homework Statement
Suppose R1 and R2 are relations on A and R1 ⊆ R2.
Let S1 and S2 be the transitive closures of R1 and R2 respectively.
Prove that S1 ⊆ S2.
Please check my proof and please explain my mistakes. thank you for taking the time to help.
Homework Equations
N/A
The Attempt at a Solution
1. Suppose R1⊆ R2 and S1 and S2 are transitive closures of R1 & R2 respectively.
2. Suppose (x,y)∈S1 such that S1=∩F where F={T⊆A x A| R1⊆T and T is transitive}
3. case1:
3.1) Suppose (x,y)∈S1 and (x,y)∉R1 such that ∃z∈A((x,z)∈R1 ∧ (z,y)∈R1) because S1 is transitive.
3.2) because R1⊆T, (x,z)∈T and (z,y)∈T. since T is transitive (x,y)∈T and S1⊆T.
3.3) since R1⊆R2, then (x,z)∈R2 and (z,y)∈R2.
3.4) let S2=∩G where G={U⊆AxA| R2⊆U and U is transitive}
3.5) since R2⊆U, (x,z)∈U, (z,y)∈U, and U is transitive then (x,y)∈U.
3.6) since U is arbitrary, ∀U∈G((x,y)∈U) then (x,y)∈∩G=S2 or (x,y)∈S2
4. case2:
4.1) Suppose (x,y)∈R1 and (x,y)∈S1
4.2) since R1⊆R2 then (x,y)∈R2
4.3) since R2⊆S2 then (x,y)∈S2
5. Therefore, S1⊆S2.
QED