associativity_checker.sagews

A binary operation on $A$ is a function $f$ such that $f:A \times A \rightarrow A$.

typeset_mode(True, display=False)

a,b,c,d, e, f = var('a b c d e f')

def ex_1_d(tup1=(a, b), tup2=(c,d)):
    return (tup1[0]*tup2[1]+tup1[1]*tup2[0], tup1[1]*tup2[1])

LHS = ex_1_d((a,b), ex_1_d((b,c), (e,f)))
print RHS == LHS
show (RHS)
show ("(", RHS[0].expand(),",", RHS[1].expand(), ")")
show ("(", LHS[0].expand(),",", LHS[1].expand() , ")")

True

($\displaystyle b c e + {\left(b^{2} + a c\right)} f$, $\displaystyle b c f$)

( $\displaystyle b c e + b^{2} f + a c f$ , $\displaystyle b c f$ )

( $\displaystyle b c e + b^{2} f + a c f$ , $\displaystyle b c f$ )

def ex_1_b(a,b):
    return a + b + a*b
RHS = ex_1_b(ex_1_b(a,b),d)
LHS = ex_1_b(a, ex_1_b(b, d))
(RHS-LHS).expand()

$0$

def ex_1_c(a,b):
    return (a+b)/5
RHS = ex_1_c(ex_1_b(a,b),c)
LHS = ex_1_c(a, ex_1_b(b, c))
(RHS-LHS).expand()

$\frac{1}{5} \, a b - \frac{1}{5} \, b c$

def ex_1_d(a,b):
    return a/b

RHS = ex_1_d(ex_1_d(a,b),d)
LHS = ex_1_d(a, ex_1_d(b, d))
RHS == LHS

$\frac{a}{b d} = \frac{a d}{b}$