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)
\(a*b=\left( \frac { a-1 }{ b-1 } \right) ,\forall a,b\in Q\)

Verify the
(i) closure property,
(ii) commutative property,
(iii) associative property
(iv) existence of identity and
(v) existence of inverse for the arithmetic operation + on Z.

Let A =\(\begin{bmatrix} 0 & 1 \\ 1 & 1 \end{bmatrix},B=\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\)be any two boolean matrices of the same type. Find AvB and A\(\wedge\)B.

Let p: Jupiter is a planet and q: India is an island be any two simple statements. Give verbal sentence describing each of the following statements.
(i) ¬p
(ii) p ∧ ¬q
(iii) ¬p ∨ q
(iv) p➝ ¬q
(v) p↔q

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

Fill in the following table so that the binary operation ∗ on A = {a, b, c} is commutative.

*	a	b	c
a	b
b	c	b	a
c	a		c

Write the converse, inverse, and contrapositive of each of the following implication.
If x and y are numbers such that x = y, then x² = y²

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

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

Construct the truth table for the following statements.
(¬p ⟶ r) ∧ ( p ↔️ q)

Verify whether the following compound propositions are tautologies or contradictions or contingency
(p ∧ q) ∧ ¬ (p ∨ q)

Verify whether the following compound propositions are tautologies or contradictions or contingency
(( p V q)∧ ¬ p) ➝ q

Verify whether the following compound propositions are tautologies or contradictions or contingency
( p ⟶ q) ↔️ (~ p ⟶ q)

Verify whether the following compound propositions are tautologies or contradictions or contingency
((p⟶ q) ∧ (q ⟶ r)) ⟶ (p ⟶ r)

Show that ¬( p ∧ q) ≡ ¬p V ¬q

Prove that q ➝ p ≡ ¬p ➝ ¬q

Show that p ➝ q and q ➝ p are not equivalent

Using truth table check whether the statements ¬(p V q) V (¬p ∧ q) and ¬p are logically equivalent.

Prove p⟶(q⟶r) ☰ (p ∧ q)⟶r without using truth table.