Examine the binary operation (closure property) of the following operations on the respective sets (if it is not, make it binary)
a*b = a + 3ab − 5b²; ∀a,b∈Z

Examine the binary operation (closure property) of the following operations on the respective sets (if it is not, make it binary)
\(a*b=\left( \frac { a-1 }{ b-1 } \right) ,\forall a,b\in Q\)

How many rows are needed for following statement formulae?
\(p \vee \neg t \wedge(p \vee \neg s)\)

Determine whether ∗ is a binary operation on the sets given below.
(a*b) = a√b is binary on R

Let A = {a +\(\sqrt5\) b : a,b∈Z}. Check whether the usual multiplication is a binary operation on A.

Determine the truth value of each of the following statements
(i) If 6 + 2 = 5 , then the milk is white.
(ii) China is in Europe or \(\sqrt3\) is an integer
(iii) It is not true that 5 + 5 = 9 or Earth is a planet
(iv) 11 is a prime number and all the sides of a rectangle are equal

Construct the truth table for the following statements.
​​​​​​¬p ∧ ¬q

Construct the truth table for the following statements.
​​​​​​¬(p ∧ ¬q)

Construct the truth table for the following statements.
( p V q) V ¬q

Let \(A=\left( \begin{matrix} 1 & 0 \\ 0 & 1 \\ 1 & 0 \end{matrix}\begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 1 \end{matrix} \right) ,B=\left( \begin{matrix} 0 & 1 \\ 1 & 0 \\ 1 & 0 \end{matrix}\begin{matrix} 0 & 1 \\ 1 & 0 \\ 0 & 1 \end{matrix} \right) ,C=\left( \begin{matrix} 1 & 1 \\ 0 & 1 \\ 1 & 1 \end{matrix}\begin{matrix} 0 & 1 \\ 1 & 0 \\ 1 & 1 \end{matrix} \right) \)be any three boolean matrices of the same type.
Find (A∨B)∧C 

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.