Invertible Element Binary Operation

Q.

The identity element of subtraction is

Q. If $a\ast b=\frac{ab}{10};a,b\in Q^{+}$, then $(5\ast 8{)}^{-1}=$______

1. $4$
2. $\frac{1}{25}$
3. $25$
4. $10$

Q.

Find. $30-[15-\left\{8-(6-8-7)\right\}]+14÷7$

Q.

Find the value of : $1+(-475)+(-475)+(-475)+(-475)+1900$

Q.

Given a non-empty set X, let *: P(X) × P(X) → P(X) be defined as A * B = (A − B) ∪ (B − A), &mnForE; A, B ∈ P(X). Show that the empty set Φ is the identity for the operation * and all the elements A of P(X) are invertible with A⁻¹ = A. (Hint: (A − Φ) ∪ (Φ − A) = A and (A − A) ∪ (A − A) = A * A = Φ).

Q. Let $A=Q\times Q$ and let $\ast$ be a binary operation on $A$ defined by $(a,b)\ast (c,d)=(ac,b+ad)$ for $(a,b)(c,d)ϵA$. Determine, whether $\ast$ is commutative and associative. Then, with respect to $\ast$ on $A$

$\left(i\right)$ Find the identity element in $A$

$\left(ii\right)$ Find the invertible elements of $A$.

Q.

Using properties of determinants, prove that:

Q.

Match the following

Column 1	Column 2
$425\times 136=425\times (6+30+100)$	Commutativity under multiplication.
$2\times 49\times 50=2\times 50\times 49$	Commutativity under addition.
$80+2005+20=80+20+2005$	Distributivity of multiplication over addition.

Q.

Evaluate P (A ∪ B), if 2P (A) = P (B) =and P(A|B) =

Q. m is said to be related to n if m and n are integers and m − n is divisible by 13. Does this define an equivalence relation?

Q.

It is given that $f\left(1\right)+2f\left(2\right)+...+nf\left(n\right)=n(n+1)f\left(n\right)$.The value of$f\left(1\right)=1$and $f\left(999\right)$is$\frac{1}{k}$ , where $k$ equals.

Q.

If X and Y are two sets such that X has 40 elements, X ∪Y has 60 elements and X ∩Y has 10 elements, how many elements does Y have?

Q. Prove the following identities:

$\left|\begin{array}{ccc}{a}^3& 2& a\\ b^3& 2& b\\ c^3& 2& c\end{array}\right|=2\left(a-b\right)\left(b-c\right)\left(c-a\right)\left(a+b+c\right)\begin{array}{}\\ \\ \end{array}$

Q. If A and B are invertible matrices, which of the following statement is not correct.
(a) $\mathrm{adj}A=\left|A\right|A^{-1}$
(b) $\det \left(A^{-1}\right)={\left(\det A\right)}^{-1}$
(c) ${\left(A+B\right)}^{-1}=A^{-1}+B^{-1}$
(d) ${\left(AB\right)}^{-1}=B^{-1}{A}^{-1}$

Q.

Using properties of determinants, prove that:

Q.

Define a binary operation *on the set {0, 1, 2, 3, 4, 5} as

Show that zero is the identity for this operation and each element a ≠ 0 of the set is invertible with 6 − a being the inverse of a.

Q. Determine whether or not each of the definition of * given below gives a binary operation. In the event that * is not a binary operation give justification of this.
(i) On Z⁺, defined * by a * b = a − b
(ii) On Z⁺, defined * by a * b = ab
(iii) On R, define by a*b = ab²
(iv) On Z⁺ define * by a * b = |a − b|
(v) On Z⁺, define * by a * b = a
(vi) On R, define * by a * b = a + 4b²
Here, Z⁺ denotes the set of all non-negative integers.

Q. Mark the correct alternative in the following question:

For the binary operation * on Z defined by a * b = a + b + 1, the identity clement is

(a) 0 (b) $-$1 (c) 1 (d) 2

Q. On the set Q⁺ of all positive rational numbers a binary operation * is defined by $a*b=\frac{ab}{2}\mathrm{for}\mathrm{all},a,b\in Q^{+}$. The inverse of 8 is
(a) $\frac{1}{8}$

(b) $\frac{1}{2}$

(c) 2

(d) 4

Q. Let * be a binary operation on Q₀ (set of non-zero rational numbers) defined by
$a*b=\frac{ab}{5}\mathrm{for}\mathrm{all}a,b\in Q_0$.
Show that * is commutative as well as associative. Also, find its identity element if it exists.

Q.

Given a non-empty set X, consider the binary operation *: P(X) × P(X) → P(X) given by A * B = A ∩ B &mnForE; A, B in P(X) is the power set of X. Show that X is the identity element for this operation and X is the only invertible element in P(X) with respect to the operation*.

Q. Let * be a binary operation on Q − {−1} defined by
a * b = a + b + ab for all a, b ∈ Q − {−1}
Then,
(i) Show that '*' is both commutative and associative on Q − {−1}.
(ii) Find the identity element in Q − {−1}
(iii) Show that every element of Q − {−1} is invertible. Also, find the inverse of an arbitrary element.

Q. Consider the binary operation * defined on Q − {1} by the rule
a * b = a + b − ab for all a, b ∈ Q − {1}
The identity element in Q − {1} is
(a) 0
(b) 1
(c) $\frac{1}{2}$
(d) −1

Q. $Let\text{'}*\text{'}beabinaryoperationonsetQ-\left\{1\right\}definedbya*b=a+b-abforalla,b\in Q-\left\{1\right\}.$
Then, which of the following statement(s) is/are true?

1. 0 is the identity element with respect to * on $Q-${1}.
2. Every element of $Q-$ {1} is invertible.
3. For any element $a\in Q-${1}, inverse of a is $\frac{a}{a-1}$
4. * is associative on $Q-$ {1}

Q. Let A = R₀ × R, where R₀ denote the set of all non-zero real numbers. A binary operation '⊙' is defined on A as follows :
(a, b) ⊙ (c, d) = (ac, bc + d) for all (a, b), (c, d) ∈ R₀ × R.
(i) Show that '⊙' is commutative and associative on A
(ii) Find the identity element in A
(iii) Find the invertible elements in A.

Q. Let $A=Q\times Q$ and let $\ast$ be a binary operation on A defined by $(a,b)\ast (c,d)=(ac,b+ad)$ for $(a,b),(c,d)\in A$. Determine, whether $\ast$ is commutative and associative. Then, with respect to $\ast$ on $A$.
(i) Find the identify element in A.
(ii) Find the invertible element of A.

Q. Let * be a binary operation defined on set Q − {1} by the rule a * b = a + b −ab. Then, the identify element for * is
(a) 1

(b) $\frac{a-1}{a}$

(c) $\frac{a}{a-1}$

(d) 0

Q. Let R₀ denote the set of all non-zero real numbers and let A = R₀ × R₀. If '*' is a binary operation on A defined by
(a, b) * (c, d) = (ac, bd) for all (a, b), (c, d) ∈ A
(i) Show that '*' is both commutative and associative on A
(ii) Find the identity element in A
(iii) Find the invertible element in A.

Q. Given a non-empty set X , let *: P( X ) × P( X ) → P( X ) be defined as A * B = ( A − B ) ∪ ( B − A ), &mnForE; A , B ∈ P( X ). Show that the empty set Φ is the identity for the operation * and all the elements A of P( X ) are invertible with A −1 = A . (Hint: ( A − Φ ) ∪ ( Φ − A ) = A and ( A − A ) ∪ ( A − A ) = A * A = Φ ).

Q. A binary operation * is defined on the set R of all real numbers by the rule

$a*b=\sqrt{a^2+b^2}\mathrm{for}\mathrm{all}a,b\in R.$

Write the identity element for * on R.