Properties of operators and functions:

• Set difference is not commutative
• Union,intersection,symmetric difference are commutative and associative
• Union and intersection are distributive over each other
• Compliment is a involution
• Squaring can lead to extraneous roots, dividing can lead to loss of roots
• Sigma(Σ) is a linear operator, it is not distributive over multiplication or division
• Factorial(!) is not distributive over addition or multiplication
• $n!|\prod \left(k\text{ consecutive whole numbers}\right)$
• $\text{If }p\text{ is prime then }p|\left[p,r\right],\forall r\in N,r<p$

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Properties of non-functions

• $\frac{f\left(x\right)}{x-k}\text{ has remainder of }f\left(k\right)$
• Odd degree polynomials have at least 1 real root
• Roots with multiplicity of n exist as roots in the first n−1 derivatives
• Between 2 roots of a polynomial, there exists at least 1 root of the derivative of the polynomial
• And rule and or rule are applicable for any number of tasks