Discrete Mathematics
Discrete Mathematics tests binary operations and algebraic structures: semigroups, monoids, groups, rings and fields. The exam often defines an unusual operation and asks you to verify axioms or find an identity and inverse.
Axioms
A binary operation * on a set S is closed if ab∈S for all a,b. It is associative if (ab)c=a(bc). An identity e satisfies ae=ea=a. An inverse b of a satisfies ab=ba=e. A group is a set with an associative operation, an identity, and inverses for every element; it is abelian if ab=b*a. A semigroup has closure and associativity; a monoid adds an identity. A ring is an abelian group under addition and a semigroup under multiplication, with distributivity. A field is a commutative ring with unity in which every nonzero element has a multiplicative inverse.
Worked example
On the integers, define ab=a+b+1. Closure holds. Associativity: (ab)c=a+b+1+c+1=a+b+c+2=a(bc). Identity: ae=a gives a+e+1=a, so e=−1. Inverse: a*b=−1 gives a+b+1=−1, so b=−a−2. The operation is commutative, so the integers form an abelian group under *.
Traps
Check closure inside the given set; an operation can be associative on integers but not on a subset. The identity is not always 0 or 1-it depends on the operation. The inverse is with respect to the defined identity. A group need not be abelian, and subtraction is not associative. For a field, every nonzero element must have an inverse, not just some.