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.

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.

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_e = the set of all even integers

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_o = the set of all odd integers

Verify 
(i) closure property  
(ii) commutative property, and 
(iii) associative property of the following operation on the given set. (a*b) = a^b;∀a, b∈N (exponentiation property)

Verify 
(i) closure property 
(ii) commutative property
(iii) associative property
(iv) existence of identity, and
(v) existence of inverse for following operation on the given set m*n = m + n - mn; m, n ∈Z

Verify 
(i) closure property 
(ii) commutative property 
(iii) associative property 
(iv) existence of identity and
(v) existence of inverse for the operation +₅ on Z₅ using table corresponding to addition modulo 5.

Verify 
(i) closure property 
(ii) commutative property 
(iii) associative property
(iv) existence of identity and
(v) existence of inverse for the operation ×11 on a subset A = {1, 3, 4, 5, 9} of the set of remainders {0,1, 2, 3, 4, 5, 6, 7, 8, 9,10}

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

Define an operation \(*\)on Q as follows: a * b =\(\left( \frac { a+b }{ 2 } \right) \); a,b ∈Q. Examine the closure, commutative, and associative properties satisfied by \(*\)on Q.

Define an operation∗ on Q as follows:  a*b = \(\left( \frac { a+b }{ 2 } \right) \); a,b ∈Q. Examine the existence of identity and the existence of inverse for the operation * on 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)

Let M = \(\left\{ \left( \begin{matrix} x & x \\ x & x \end{matrix} \right) :x\in R-\{ 0\} \right\} \) and let * be the matrix multiplication. Determine whether M is closed under ∗. If so, examine the commutative and associative properties satisfied by ∗ on M.

Let M = \(\left\{ \left( \begin{matrix} x & x \\ x & x \end{matrix} \right) :x\in R-\{ 0\} \right\} \) and let ∗ be the matrix multiplication. Determine whether M is closed under ∗ . If so, examine the existence of identity, existence of inverse properties for the operation ∗ on M.

Let A be Q\{1}. Define ∗ on A by x*y = x + y − xy. Is ∗ binary on A? If so, examine the commutative and associative properties satisfied by ∗ on A.

Check whether the statement p➝(q➝p) is a tautology or a contradiction without using the truth table.

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

Let A be Q\{1}. Define ∗ on A by x*y = x + y − xy . Is ∗ binary on A? If so, examine the existence of identity, existence of inverse properties for the operation ∗ on A.

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

Prove that p➝(¬q V r) ≡ ¬pV(¬qVr) using truth table.

Show that \(\neg(p \wedge q) \equiv \neg p \vee \neg q\)

Show that \(\neg(p \rightarrow q) \equiv p \wedge \neg q\)