# More Mathematical Logic

Axioms of equality are basic rules for using the equals sign…

If a=b and c=d, then a+c=b+d.

If a>b, then a+c>b+c.

Any number added to 0 gives the original number, for instance, n+0=n.

Identity Element For Multiplication:

Any number multiplied by 1 gives the original number, for instance, n*1=n.

Multiplicative Axiom Of Equality:

If a=b and c=d, then ac=bd.

Negative Multiplication Property Of Inequality:

You must reverse the inequality sign when multiplying or dividing by a negative number.

# How To Expand (a+b+c)(d+e+f)(g+h+i)

Firstly you have to know what (a+b+c)(d+e+f) is. You can expand this expression using a rectangle:

So you know that: $\left( a+b+c \right) \left( d+e+f \right) =ad+ae+af+bd+be+bf+cd+ce+cf$

Next you’d have to multiply (a+b+c)(d+e+f) by (g+h+i) using another rectangle:

And from here you’d figure out that:

$\left( a+b+c \right) \left( d+e+f \right) \left( g+h+i \right) \\ \\ =\left[ \left( a+b+c \right) \left( d+e+f \right) \right] \left( g+h+i \right) \\ \\ =\left[ ad+ae+af+bd+be+bf+cd+ce+cf \right] \left( g+h+i \right) \\ \\ =adg+aeg+afg+bdg+beg+bfg+cdg+ceg+cfg\\ \\ +adh+aeh+afh+bdh+beh+bfh+cdh+ceh+cfh\\ \\ +adi+aei+afi+bdi+bei+bfi+cdi+cei+cfi\\ \\ =ag\left( d+e+f \right) +bg\left( d+e+f \right) +cg\left( d+e+f \right) \\ \\ +ah\left( d+e+f \right) +bh\left( d+e+f \right) +ch\left( d+e+f \right) \\ \\ +ai\left( d+e+f \right) +bi\left( d+e+f \right) +ci\left( d+e+f \right) \\ \\ =\left( d+e+f \right) \left[ ag+bg+cg+ah+bh+ch+ai+bi+ci \right] \\ \\ =\left( d+e+f \right) \left[ g\left( a+b+c \right) +h\left( a+b+c \right) +i\left( a+b+c \right) \right] \\ \\ =\left( d+e+f \right) \left[ \left( a+b+c \right) \left( g+h+i \right) \right] \\ \\ =\left( a+b+c \right) \left( d+e+f \right) \left( g+h+i \right)$