# Invertible Element Binary Operation

## Trending Questions

**Q.**

The identity element of subtraction is

**Q.**If a∗b=ab10; a, b∈Q+, then (5∗8)−1=______

- 4
- 125
- 25
- 10

**Q.**

Find. $30-[15-\left\{8-(6-8-7)\right\}]+14\xf77$

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

(i) Find the identity element in A

(ii) 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) $\mathrm{det}\left({A}^{-1}\right)={\left(\mathrm{det}A\right)}^{-1}$

(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 = ab

^{2}

(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 + 4b

^{2}

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 Q

_{0}(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 ' * ' be a binary operation on set Q − {1} defined bya * b = a + b − ab for all a, b ∈ Q − {1}.

Then, which of the following statement(s) is/are true?

- 0 is the identity element with respect to * on Q−{1}.
- Every element of Q− {1} is invertible.
- For any element a∈Q−{1}, inverse of a is aa−1
- * is associative on Q− {1}

**Q.**Let A = R

_{0}× R, where R

_{0}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) ∈ R

_{0}× 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×Q and let ∗ be a binary operation on A defined by (a, b)∗(c, d)=(ac, b+ad) for (a, b), (c, d)∈A. Determine, whether ∗ is commutative and associative. Then, with respect to ∗ 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 R

_{0}denote the set of all non-zero real numbers and let A = R

_{0}× R

_{0}. 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.