Discrete Math with SageMath: Learn math with open-source software

If we have two relations with a set in common, such as $R$ from $A$ to $B$ and $S$ from $B$ to $C\text{,}$ we can compose them to create a relation between the two non repeated sets, such as $S\circ R$ from $A$ to $C\text{,}$ where the relation is defined as all $aS\circ Rc$ where there exists an element $b\in B$ such that $aRb$ and $bSc$ exist.

We can also use Sage to compose relations.

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A = Set([1, 2, 3, 4, 5, 6])

B = Set([1, 2, 7])

R = Set([(a, b) for a in A for b in B if a==2*b])

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C = Set([7,8,9,10])

S = Set([(a,c) for a in A for c in C if a + c == 10])

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SoR = Set([(r[0], s[1]) for r in R for s in S if r[1] == s[0]])

show(SoR)

When $A=B$ we refer to the relation as a relation on $A\text{.}$

Consider the set $A=\{2,3,4,6,8\}\text{.}$ Let’s define a relation $R$ on $A$ such that $aRb$ iff $a|b$ ($a$ divides $b$). The relation $R$ can be represented by the set of ordered pairs where the first element divides the second:

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A = Set([2, 3, 4, 6, 8])

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# Define the relation R on A: aRb iff a divides b

R = Set([(a, b) for a in A for b in A if a.divides(b)])

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show(R)