List any four rules of inference. Using direct and indirect proof show that, for any real number $x$, if $x^3 - 7x^2 + x - 7 = 0$, then $x = 7$[10]

What are the necessary and sufficient conditions for graphs to have Euler path only and Euler circuit? Let $R$ be a relation defined on set of natural numbers $N$, such that $a,b \in N, aRb \leftrightarrow ab = 2k$, where $k \in \{0,1,2,...\}$. Show that $R$ is a partial ordering relation on $N$[10]

Determine whether the function f(x) = x-1 from set of integers to the set of integers is injective, surjective or bijective. Write the type of Fuzzy set operations with their definition. Give the floor and ceiling value of 1.2 and -1.2.[10]