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Question 6. Let ∗ be the binary operation on N given by a ∗ b = L.C.M. of a and b. Find

(v) Which elements of N are invertible for the operation ∗?

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Q12  Let A = {9,10,11,12,13} and let f : A\rightarrow N be defined by f (n) = the highest prime
factor of n. Find the range of f.

It is given that   A = {9,10,11,12,13} And  f : A → N be defined by f(n) = the highest prime factor of n. Now, Prime factor of 9 = 3 Prime factor of 10 = 2,5 Prime factor of 11 = 11 Prime factor of 12 = 2,3 Prime factor of 13 = 13 f(n) = the highest prime factor of n. Hence, f(9) = the highest prime factor of 9 = 3 f(10) = the highest prime factor of 10 = 5 f(11) = the highest prime factor...

Q11  Let f be the subset of Z \times Z defined by f = \left \{ (ab, a + b) : a, b \epsilon Z } \right \} . Is f a
         function from Z to Z? Justify your answer.

It is given that  Now, we know that relation f from a set A to a set B is said to be a function only if every element of set A has a unique image in set B  Now,  for value 2, 6, -2, -6  Now,  we can observe that same first element i.e. 12 corresponds to two different images that are 8 and -8. Thus, f   is not a function

Q10 (2)  Let A ={1,2,3,4}, B = {1,5,9,11,15,16} and f = {(1,5), (2,9), (3,1), (4,5), (2,11)} Are the following true?  f is a  function from A to B

It is given that      and  Now, As we can observe that same first element i.e. 2 corresponds to two different images that is 9 and 11. Hence f is not a function from A to B Therefore, given statement is FALSE

Q10 (1)  Let A ={1,2,3,4}, B = {1,5,9,11,15,16} and f = {(1,5), (2,9), (3,1), (4,5), (2,11)}
               Are the following true?  f is a relation from A to B

It is given that      and  Now, Now, a relation from a non-empty set A to a non-empty set B is a subset of the Cartesian product A × B And we can see that f is a subset of  Hence  f is a relation from A to B  Therefore, given statement is TRUE

Q9 (3)  Let R be a relation from N to N defined by R = \left \{ ( a,b): a,b \epsilon N \: \:and \: \: a = b ^ 2 \right \} . Are
            the following true?

       (a,b) \epsilon R, (b,c) \epsilon R implies (a,c) \epsilon R.

It is given that And   implies  Now, it can be seen that    because    and  ,  But   Therefore,   Therefore, the given statement is FALSE  

Q9 (2)  Let R be a relation from N to N defined by R = \left \{ ( a,b): a,b \epsilon N \: \:and \: \: a = b ^ 2 \right \} . Are
            the following true?

            ( a,a ) \epsilon R , implies (b,a)  \epsilon R

It is given that And   implies  Now , it can be seen that    and  ,  But   Therefore,   Therefore,  given statement is FALSE  

Q9 (1)  Let R be a relation from N to N defined by R = \left \{ ( a,b): a,b \epsilon N \: \:and \: \: a = b ^ 2 \right \} . Are
            the following true?

            ( a,a ) \epsilon R , for all a \epsilon N

It is given that And     for all    Now, it can be seen that    But,  Therefore, this statement is FALSE 

Q8  Let f = {(1,1), (2,3), (0,–1), (–1, –3)} be a function from Z to Z defined by
       f(x) = ax + b, for some integers a, b. Determine a, b.

It is given that  And  Now, At x = 1 ,  Similarly, At  ,  Now, put this value of  b  in equation (i) we will get, Therefore, values of a and  b are 2 and -1  respectively

Q7 Let f, g : R \rightarrow R be defined, respectively by f(x) = x + 1, g(x) = 2x – 3. Find
       f + g, f – g and f/g

It is given that Now,                                           Therefore, Now,                                                                Therefore, Now,                  Therefore, values of   are    respectively    

Q6  Let

  f = \left \{ \left ( x , \frac{x^2}{1+ x^2} \right ) : x \epsilon R \right \} 
R be a function from R into R. Determine the range
of f.

Given function is  Range of any function is the set of values obtained after the mapping is done in the domain of the function. So every value of the codomain that is being mapped is Range of the function. Let's take   Now, 1 - y should be greater than zero and y should be greater than and equal to zero for x to exist because other than those values the x will be...

Q5 Find the domain and the range of the real function f defined by f (x) = |x-1|

Given function is  As the given function is defined of all real number  The domain of the function   is  R Now, as we know that the mod function always gives only  positive values Therefore, Range  of function   is  all non-negative real numbers i.e. 

Q4  Find the domain and the range of the real function f defined by f (x) = \sqrt{(x-1)}

Given function is  We can clearly see that f(x) is only defined for the values of x ,   Therefore, The domain of the function   is   Now, as take square root on both sides Therefore, Range  of function   is  

3. Find the domain of the function f (x) = \frac{x^2 + 2 x +1}{x^2 - 8x + 12 }

Given function is  Now, we will simplify it into                                     Now, we can clearly see that   Therefore, the Domain of f(x)  is   

Q2  If f (x)= x^2  find \frac{f ( 1.1)- f (1)}{(1.1-1)}

Given function is  Now, Therefore, value of   is  2.1

1. The relation f is defined by
f (x) = \left\{\begin{matrix} x^2 & 0 \leq x\leq 3 \\ 3x &3 \leq x \leq 10 \end{matrix}\right.
The relation g is defined by
g(x) = \left\{\begin{matrix} x^2 & 0 \leq x\leq 2 \\ 3x &2 \leq x \leq 10 \end{matrix}\right.
Show that f is a function and g is not a function.

It is given that  Now, And At x = 3, Also, at x = 3, We can see that for , f(x) has unique images. Therefore, By definition of a function, the given relation is function. Now, It is given that Now, And At x = 2, Also, at x = 2, We can clearly see that element 2 of the domain of relation g(x) corresponds to two different images i.e. 4 and 6. Thus,  f(x) does not have unique...

Q5 (3)  Find the range of each of the following functions.

            f (x) = x, x is a real number

Given function is  It is given that  x is a real  number Therefore, Range of function   is  R

Q5 (2)  Find the range of each of the following functions

         f ( x ) = x ^2 +2 , x is a real number.

Given function is  It is given that  x is a real  number Now, Add 2 on both the sides  Therefore, Range of function  is  

Q5 (1)  Find the range of each of the following functions.

         f (x) = 2 - 3x, x \epsilon R, x > 0.

Given function is  It is given that  Now, Add 2 on both the sides  Therefore, Range of function  is  

Q4(4)    The function ‘t’ which maps temperature in degree Celsius into temperature in
       degree Fahrenheit is defined by t ( C ) = \frac{9 C }{5} + 32
      The value of C, when t(C) = 212.

Given function is  Now, Therefore, When t(C) = 212 , value of C is  100 
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