Show that p v (~p) is a tautology.

Show that p v (q ∧ r) is a contingency.

In the set of integers under the operation * defined by a * b = a + b - 1. Find the identity element.

Let S be the set of positive rational numbers and is defined by a * b = \(\frac{ab}{2}\). Then find the identity element and the inverse of 2.

Let G = {1, w, w²) where w is a complex cube root of unity. Then find the universe of w². Under usual multiplication.

Let * be a binary operation on the set of all nonzero real numbers, given by  \(\mathrm{a} * \mathrm{b}=\frac{a b}{5} \forall a, b \in R\). Find the value of x, given that \(2 *(x * 5)=10\)

Show that subtraction and division are not binary operations on N.

Form the truth table for \(\neg[p \vee(\neg q)]\)

Show that \(p \wedge q \rightarrow p\) is a tautology.

Let * be a binary operation on set Q of rational numbers defind as \(a * b=\frac{a b}{8}\). Write the identity for *, if any.