Advertisements
Advertisements
Question
Let +6 (addition modulo 6) be a binary operation on S = {0, 1, 2, 3, 4, 5}. Write the value of \[2 +_6 4^{- 1} +_6 3^{- 1} .\]
Solution
Here,
1 \[+_6\] 1 = Remainder obtained by dividing 1 + 1 by 6
= 2
3 \[+_6\] 4 = Remainder obtained by dividing 3 + 4 by 6
= 1
4 \[+_6\] 5 = Remainder obtained by dividing 4 + 5 by 6
= 3
So, the composition table is as follows:
| +6 | 0 | 1 | 2 | 3 | 4 | 5 |
| 0 | 0 | 1 | 2 | 3 | 4 | 5 |
| 1 | 1 | 2 | 3 | 4 | 5 | 0 |
| 2 | 2 | 3 | 4 | 5 | 0 | 1 |
| 3 | 3 | 4 | 5 | 0 | 1 | 2 |
| 4 | 4 | 5 | 0 | 1 | 2 | 3 |
| 5 | 5 | 0 | 1 | 2 | 3 | 4 |
We observe that the first row of the composition table coincides with the the top-most row and the first column coincides with the left-most column.
These two intersect at 0.
\[\Rightarrow a +_6 0 = 0 +_6 a = a, \forall a \in S\]
So, 0 is the identity element.
From the table,
\[4 +_6 2 = 0 \]
\[ \Rightarrow 4^{- 1} = 2\]
\[3 +_6 3 = 0 \]
\[ \Rightarrow 3^{- 1} = 3\]
\[\text{Now},\]
\[2 +_6 4^{- 1} +_6 3^{- 1} = 2 +_6 2 +_6 3\]
\[ = 4 +_6 3 \]
\[ = 1\]
APPEARS IN
RELATED QUESTIONS
Show that the binary operation * on A = R – { – 1} defined as a*b = a + b + ab for all a, b ∈ A is commutative and associative on A. Also find the identity element of * in A and prove that every element of A is invertible.
Let A = Q ✕ Q, where Q is the set of all rational numbers, and * be a binary operation defined on A by (a, b) * (c, d) = (ac, b + ad), for all (a, b) (c, d) ∈ A.
Find
(i) the identity element in A
(ii) the invertible element of A.
(iii)and hence write the inverse of elements (5, 3) and (1/2,4)
Determine whether or not of the definition of given below gives a binary operation. In the event that * is not a binary operation, give justification for this.
On Z+, define * by a * b = ab
For each binary operation * defined below, determine whether * is commutative or associative.
On Z, define a * b = a − b
For each binary operation * defined below, determine whether * is commutative or associative.
On Q, define a * b = ab + 1
For each binary operation * defined below, determine whether * is commutative or associative.
On R − {−1}, define `a*b = a/(b+1)`
Let*′ be the binary operation on the set {1, 2, 3, 4, 5} defined by a *′ b = H.C.F. of a and b. Is the operation *′ same as the operation * defined in Exercise 4 above? Justify your answer.
Is * defined on the set {1, 2, 3, 4, 5} by a * b = L.C.M. of a and b a binary operation? Justify your answer.
Define a binary operation *on the set {0, 1, 2, 3, 4, 5} as
a * b = `{(a+b, "if a+b < 6"), (a + b - 6, if a +b >= 6):}`
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.
Let A = Q x 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
1) Find the identity element in A
2) Find the invertible elements of A.
If a * b denotes the larger of 'a' and 'b' and if a∘b = (a * b) + 3, then write the value of (5)∘(10), where * and ∘ are binary operations.
Determine whether the following operation define a binary operation on the given set or not : '*' on N defined by a * b = ab for all a, b ∈ N.
Determine whether the following operation define a binary operation on the given set or not : '*' on N defined by a * b = a + b - 2 for all a, b ∈ N
Determine whether or not the definition of *given below gives a binary operation. In the event that * is not a binary operation give justification of this.
On Z+, defined * by a * b = ab
Here, Z+ denotes the set of all non-negative integers.
Let * be a binary operation on the set I of integers, defined by a * b = 2a + b − 3. Find the value of 3 * 4.
Is * defined on the set {1, 2, 3, 4, 5} by a * b = LCM of a and b a binary operation? Justify your answer.
Check the commutativity and associativity of the following binary operation'*' on Q defined by a * b = ab + 1 for all a, b ∈ Q ?
Check the commutativity and associativity of the following binary operation '*' on Z defined by a * b = a + b − ab for all a, b ∈ Z ?
Show that the binary operation * on Z defined by a * b = 3a + 7b is not commutative ?
On the set Z of integers a binary operation * is defined by a * b = ab + 1 for all a , b ∈ Z. Prove that * is not associative on Z.
On Z, the set of all integers, a binary operation * is defined by a * b = a + 3b − 4. Prove that * is neither commutative nor associative on Z.
The binary operation * is defined by \[a * b = \frac{ab}{7}\] on the set Q of all rational numbers. Show that * is associative.
If the binary operation * on the set Z is defined by a * b = a + b −5, the find the identity element with respect to *.
On the set Z of integers, if the binary operation * is defined by a * b = a + b + 2, then find the identity element.
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 :
Find the identity element in A ?
Define a binary operation * on the set {0, 1, 2, 3, 4, 5} as \[a * b = \begin{cases}a + b & ,\text{ if a + b} < 6 \\ a + b - 6 & , \text{if a + b} \geq 6\end{cases}\]
Show that 0 is the identity for this operation and each element a ≠ 0 of the set is invertible with 6 − a being the inverse of a.
For the binary operation multiplication modulo 10 (×10) defined on the set S = {1, 3, 7, 9}, write the inverse of 3.
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 _______________ .
On the set Q+ of all positive rational numbers a binary operation * is defined by \[a * b = \frac{ab}{2} \text{ for all, a, b }\in Q^+\]. The inverse of 8 is _________ .
The number of commutative binary operations that can be defined on a set of 2 elements is ____________ .
Examine whether the operation *defined on R by a * b = ab + 1 is (i) a binary or not. (ii) if a binary operation, is it associative or not?
Let * be an operation defined as *: R × R ⟶ R, a * b = 2a + b, a, b ∈ R. Check if * is a binary operation. If yes, find if it is associative too.
Let * be defined on R by (a * b) = a + b + ab – 7. Is * binary on R? If so, find 3 * `((-7)/15)`
Define an operation * on Q as follows: a * b = `(("a" + "b")/2)`; a, b ∈ Q. Examine the closure, commutative and associate properties satisfied by * on Q.
Choose the correct alternative:
If a * b = `sqrt("a"^2 + "b"^2)` on the real numbers then * is
Let * be the binary operation defined on Q. Find which of the following binary operations are commutative
a * b = a – b ∀ a, b ∈ Q
Let * be a binary operation on set Q of rational numbers defined as a * b `= "ab"/5`. Write the identity for * ____________.
Which of the following is not a binary operation on the indicated set?
