Advertisements
Advertisements
प्रश्न
Subtraction of integers is ___________________ .
पर्याय
commutative but no associative
commutative and associative
associative but not commutative
neither commutative nor associative
Advertisements
उत्तर
neither commutative nor associative
Subtraction of integers is not commutative
For example: If a = 1 and b = 2, then both are integers
\[1 - 2 = - 1\]
\[2 - 1 = 1 \]
\[\Rightarrow - 1 \neq 1\]
\[\therefore a - b \neq b - a \forall a, b \in Z\]
Subtraction of integers is not associative.
For example: If a = 1, b = 2, c = 3, then all are integers
\[1 - \left( 2 - 3 \right) = 1 + 1\]
\[ = 2\]
\[\left( 1 - 2 \right) - 3 = - 1 - 3\]
\[ = - 4\]
\[ \Rightarrow 2 \neq - 4\]
\[ \therefore a - \left( b - c \right) \neq \left( a - b \right) - c , \forall a, b, c \in Z\]
APPEARS IN
संबंधित प्रश्न
LetA= R × R and * be a binary operation on A defined by (a, b) * (c, d) = (a+c, b+d)
Show that * is commutative and associative. Find the identity element for * on A. Also find the inverse of every element (a, b) ε A.
For each binary operation * defined below, determine whether * is commutative or associative.
On Z, define a * b = a − b
Number of binary operations on the set {a, b} are
(A) 10
(B) 16
(C) 20
(D) 8
Determine whether the following operation define a binary operation on the given set or not : '×6' on S = {1, 2, 3, 4, 5} defined by
a ×6 b = Remainder when ab is divided by 6.
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+ define * by a * b = |a − b|
Here, Z+ denotes the set of all non-negative integers.
Check the commutativity and associativity of the following binary operations '*'. on N defined by a * b = 2ab for all a, b ∈ N ?
Check the commutativity and associativity of the following binary operations '*'. on Q defined by a * b = a − b for all a, b ∈ Q ?
On the set Q of all ration numbers if a binary operation * is defined by \[a * b = \frac{ab}{5}\] , prove that * is associative on Q.
Let S be the set of all rational numbers except 1 and * be defined on S by a * b = a + b \[-\] ab, for all a, b \[\in\] S:
Prove that * is commutative as well as associative ?
Let * be a binary operation on Q − {−1} defined by a * b = a + b + ab for all a, b ∈ Q − {−1} Find the identity element in Q − {−1} ?
Let * be a binary operation on Q − {−1} defined by a * b = a + b + ab for all a, b ∈ Q − {−1} Show that every element of Q − {−1} is invertible. Also, find the inverse of an arbitrary element ?
Write the multiplication table for the set of integers modulo 5.
Define an associative binary operation on a set.
Let * be a binary operation, on the set of all non-zero real numbers, given by \[a * b = \frac{ab}{5} \text { for all a, b } \in R - \left\{ 0 \right\}\]
Write the value of x given by 2 * (x * 5) = 10.
Let +6 (addition modulo 6) be a binary operation on S = {0, 1, 2, 3, 4, 5}. Write the value of \[2 +_6 4^{- 1} +_6 3^{- 1} .\]
Let * be a binary operation defined by a * b = 3a + 4b − 2. Find 4 * 5.
Let * be a binary operation on N given by a * b = HCF (a, b), a, b ∈ N. Write the value of 22 * 4.
Let * be a binary operation on set of integers I, defined by a * b = 2a + b − 3. Find the value of 3 * 4.
If the binary operation * on Z is defined by a * b = a2 − b2 + ab + 4, then value of (2 * 3) * 4 is ____________ .
Mark the correct alternative in the following question:-
For the binary operation * on Z defined by a * b = a + b + 1, the identity element is ________________ .
If G is the set of all matrices of the form
\[\begin{bmatrix}x & x \\ x & x\end{bmatrix}, \text{where x } \in R - \left\{ 0 \right\}\] then the identity element with respect to the multiplication of matrices as binary operation, is ______________ .
On Z an operation * is defined by a * b = a2 + b2 for all a, b ∈ Z. The operation * on Z is _______________ .
A binary operation * on Z defined by a * b = 3a + b for all a, b ∈ Z, is ________________ .
The number of commutative binary operations that can be defined on a set of 2 elements is ____________ .
Let A = {a + `sqrt(5)`b : a, b ∈ Z}. Check whether the usual multiplication is a binary operation on A
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.
Let A = `((1, 0, 1, 0),(0, 1, 0, 1),(1, 0, 0, 1))`, B = `((0, 1, 0, 1),(1, 0, 1, 0),(1, 0, 0, 1))`, C = `((1, 1, 0, 1),(0, 1, 1, 0),(1, 1, 1, 1))` be any three boolean matrices of the same type. Find A v B
Let A = `((1, 0, 1, 0),(0, 1, 0, 1),(1, 0, 0, 1))`, B = `((0, 1, 0, 1),(1, 0, 1, 0),(1, 0, 0, 1))`, C = `((1, 1, 0, 1),(0, 1, 1, 0),(1, 1, 1, 1))` be any three boolean matrices of the same type. Find A ∧ B
Let M = `{{:((x, x),(x, x)) : x ∈ "R"- {0}:}}` 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 A be Q\{1}. Define * on A by x * y = x + y – xy. Is * binary on A? If so, examine the existence of an identity, the existence of inverse properties for the operation * on A
Choose the correct alternative:
A binary operation on a set S is a function from
Let * be a binary operation on Q, defined by a * b `= (3"ab")/5` is ____________.
The binary operation * defined on set R, given by a * b `= "a+b"/2` for all a, b ∈ R is ____________.
Let A = N x N and * be the binary operation on A defined by (a, b) * (c, d) = (a + c, b + d). Then * is ____________.
Find the identity element in the set I+ of all positive integers defined by a * b = a + b for all a, b ∈ I+.
The binary operation * defined on N by a * b = a + b + ab for all a, b ∈ N is ____________.
The identity element for the binary operation * defined on Q – {0} as a * b = `"ab"/2 AA "a, b" in "Q" - {0}` is ____________.
Let * be the binary operation on N given by a * b = HCF (a, b) where, a, b ∈ N. Find the value of 22 * 4.
