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If r is a relation from R to R defined by \(r = \left[ \left( a, b \right)  a, b \in R, a – b + \sqrt{3} \right ] \) is an irrational. Then the relation r is
Hence
is an irrationals.
is reflexive.
Let
is an irrational number but
is not an irrational number.
Hence is only reflexive.
R is a relation on N given by\(R = \left\{ {\left( {x,y} \right)  4x + 3y = 20} \right\}\). Which of the following belongs to R?
is a relation on
Option 1 and Option 2 are not satisfy because .
If \(R = \left\{ {\left( {3,3} \right),\left( {6,6} \right),\left( {9,9} \right),\left( {12,12} \right),\left( {6,12} \right),\left( {3,9} \right),\left( {3,12} \right),\left( {3,6} \right)} \right\}\)is a relation on the set A\( = \left\{ {3,6,9,12} \right\}\). Then the relation is
Since
So, is refelxive.
Now but
is not Symmetric also
So, is transitive.
is reflexive and transitive.
If R is an equivalence relation on a set A, then R1 is
Inverse of an equivalence relation is also an equivalence relation. So is an equivalence relation.
If \(A = \left\{ {1,3,5,7} \right\}\) and \(B = \left\{ {1,2,3,4,5,6,7,8} \right\}\) then the number of oneone function from A to B is
and
and .
Number of oneone function from into .
The function \(f\left( x \right) = {x^2} + bx + c\), where b and c are real constants, describes
This is a quadratic equation of parabola either downward or upward. Hence it is a manyone mapping.
If f and g be real functions defined by \(f\left( x \right) = 2x + 1,g\left( x \right) = 4x – 7\), for what real numbers x, \(f\left( x \right) = g\left( x \right)\)?
2x+1=4x7
2x=8
x=4
If f be real functions, \(f\left( x \right) = {x^2} + 7\), then \(\frac{{f\left( t \right) – f\left( 5 \right)}}{{t – 5}}\) is equal t
and
, where is equal to
Find the domain of the function \(f\left( x \right) = \frac{{3x}}{{x – 9}}\).
is defined, if
Domain of
Find the domain of the function \(f\left( x \right) = \frac{1}{{\sqrt {1 – \cos x} }}\)
We know
is defined, if
Domain of
If \(\left A \right = m \) and\(\left B \right = n\), then total number of nonempty relation from A to B is
Total number of relation from to .
Total number of nonempty relation from to
If \(\left[ {{x^2}} \right] – 7\left[ x \right] + 10 = 0\), where [] denotes the greatest integer function, then,
If \(f\left( x \right) = \frac{{x – 1}}{{x + 1}}\), then
But option 2 and Option 3 is not true.
Hence, .
If \(y = f\left( x \right) = \frac{{ax – b}}{{cx – a}}\), then
If \(f\left( x \right) = \sqrt {1 + {x^2}} \), then
Domain of the function \(f(x)=\sqrt{a^2x^2}, a > 0 \) is equal to_____.
is defined , if
Domain of
Range of the function \(f\left( x \right) = 2 – \left {x – 9} \right\) is
The correct option is . Because, is defined for all .
Domain of
Range of
The domain for which the functions defined by \(f\left( x \right) = 3{x^2} – 1\) and \(g\left( x \right) = 3 + x\) are equal to
and
Since ,
Domain of the function and is
If \(\left( {x – 2,y + 5} \right) = \left( { – 2,\frac{1}{3}} \right)\) are two equal ordered pairs, then
If \(A \times B = \left\{ {\left( {a,x} \right),\left( {b,x} \right),\left( {a,y} \right),\left( {b,y} \right)} \right\}\), then
Set of first elements of ordered pairs in
Set of second elements of ordered pairs in
If \(P = \left\{ {x:x < 3,x \in N} \right\}\) and \(Q = \left\{ {x:x \le 2,x \in {N^ * }} \right\}\), then\(\left {\left( {P \cup Q} \right) \times \left( {P \cap Q} \right)} \right\) is equal to
and
or
If \(a \equiv b(\bmod n)\), then
divides
If \( a \equiv b\left( {\bmod n} \right),c \equiv d\left( {\bmod n} \right)\), then
If \(5 \equiv k(\bmod 3)\), then k is equal to
is a multiple of .
i.e., and also.
If \({\log _2}a = 4\), then a is equal to