Tag: sets and relations

Questions Related to sets and relations

If the binary operation $*$ is defined on a set of ordered pairs of real numbers as $(a, b) * (c, d) = (a \times d + b \times c, b \times d)$ and is associative, then $(1, 2) * (3, 5) * (3, 4)$ is equal to

mathematics and statistics binary operations properties of binary operations discrete mathematics sets and relations

  1. $(74,40)$

  2. $(32,40)$

  3. $(23,11)$

  4. $(7,11)$

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Correct Option: A
Explanation:

Binary operation of odered pairs ;$(a,b)\ast(c,d)=(a\times d+b\times c,b\times d)$ is associative.

$\Rightarrow (1,2)\ast(3,5)\ast(3,4)=((1,2)\ast(3,5))\ast(3,4)$
$=(1\times 5+2\times 3,2\times 5)\ast(3,4)$
$=(11,10)\ast(3,4)$
$=(74,40)$

The set of all real numbers under the usual multiplication operation is not a group since

mathematics and statistics binary operations properties of binary operations discrete mathematics sets and relations

  1. multiplication is not a binary operation

  2. multiplication is not associative

  3. identity element does not exist

  4. zero has no inverse

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Correct Option: D
Explanation:

Set of real number i.e. $(-\infty,0)\cup(0,\infty)$ is under usual multiplication operation because $0\in R$ and zero do not have an inverse i.e. it can not give ordered airs to be included in a group.

$P\wedge (q \vee r)=(p\wedge q)\vee r$. This law is known as 

mathematics and statistics binary operations properties of binary operations discrete mathematics sets and relations

  1. Commutative law

  2. Associative law

  3. De-Morgan's law

  4. Distributive law

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Correct Option: A

If * is defined on the set R of all real numbers by $a*b=\sqrt{a^2+b^2}$, find the identity element in R with respect to *.

mathematics and statistics binary operations properties of binary operations discrete mathematics sets and relations

  1. 0

  2. 1

  3. 2

  4. 3

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Correct Option: A
Explanation:

Let e be the identity element in R with respect to . Then,

$a*e=a=e*a$ for all $a\in R$
$a*e=a$ and $e*a=a$ for all $a\in R$
$\sqrt{a^2+e^2}=a$ and $\sqrt{e^2+a^2}=a$ for all $a\in R$
$a^2+e^2=a^2$ and $e^2+a^2=a^2$ for all $a\in R$
$e=0$
Hence, 0 is the identity element in R with respect to $$.

Subtraction of integers is an operation that is

mathematics and statistics binary operations properties of binary operations discrete mathematics sets and relations

  1. commutative and associative

  2. not commutative but associative

  3. neither commutative nor associative

  4. commutative but not associative.

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Correct Option: C
Explanation:

By property of integers, subtraction of integers is neither associative nor commutative.
Example
$9-3=6$
$3-6=-6\neq(9-3)$
$1-(4-8)=1-(-4)=5$
$(1-4)-8=-3-8=-11$
Hence $(a-b)-c\neq a-(b-c)$

If $A={6,7,8,9}$ and $B={0,1,3,4,5}$, then $A \cap B$ is

mathematics and statistics binary operations properties of binary operations discrete mathematics sets and relations

  1. $\phi$

  2. ${6,7,8,9}$

  3. ${1,2,3,4,5,6,7,8,9}$

  4. None of the above

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Correct Option: A
Explanation:

$A\cap B= B \cap A= $ the elements which are common in both A and B. 

Hence $A\cap B= B \cap A$ $=\phi$
Since no element is common in A and B

If $\displaystyle M\cup N=N\cup R$ and $\displaystyle M\cap  N=N\cap R$  then which of the following is necessarily true?

mathematics and statistics binary operations properties of binary operations discrete mathematics sets and relations

  1. M=N

  2. N=R

  3. M=R

  4. M=N=R

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Correct Option: C
Explanation:

We know that Union and Intersection of sets is commutative.
We see that if $ R $ is replaced by $ M $, then the relations show the commutative property being satisfied.

So, $ M = R $

A closed set with respect to some binary operation is called semi- group if

mathematics and statistics binary operations properties of binary operations discrete mathematics sets and relations

  1. $*$ is associative

  2. $*$ is commutative

  3. $*$ is anti-commutative

  4. identity element exists

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Correct Option: A
Explanation:

In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation.

Hence, $A$ is correct.

$"*"$ is said to be commutative in $A$ for all $a,b  \epsilon  A$

mathematics and statistics binary operations properties of binary operations discrete mathematics sets and relations

  1. $a+b=b+a$

  2. $a*b=b*a$

  3. $a-b=b-a$

  4. $a*b\neq b*a$

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Correct Option: B
Explanation:

This is the definition of commutativeness for all binary operators.

If $A \ast B = A \cap B$ on $P(X)$, then identify for $\ast$ is ________$(X \neq \phi)$

mathematics and statistics binary operations properties of binary operations discrete mathematics sets and relations

  1. $\phi$

  2. $X$

  3. $U$

  4. $A$

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Correct Option: B
Explanation:
Given: $A \ast B = A \cap B$ on $P(X)$
Now, $P(X)$ is a power set of $X$ which is the set of all subsets of $X$
So, $P(X)$ will have $\phi$ as its smallest set and $X$ will be the largest set.
Since, $A\ast B$ is defined on $P(X)$
So, $A \cap B\in P(X)$
So, $A\subseteq X$ and $B\subseteq X$
$\therefore A\cap X=A$ and $X\cap A=A$
$\therefore A\ast X=A$ and $X\ast A=A$
Hence, $X$ is the identity element for $\ast$.