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Let \[A = \left\{ {1,2,3,4,6} \right\}\] and R be the relation on A defined by $\left\{ {(a,b):a,b \in A,b{\text{ }}is{\text{ }}exactly{\text{ }}divisible{\text{ }}by{\text{ }}a} \right\}$
A) Write R in roster form
B) Find the domain of R
C) Find the range of R

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Last updated date: 19th Jun 2024
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Answer
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Hint: Here the given set is \[A = \left\{ {1,2,3,4,6} \right\}\] and set R is defined as $\left\{ {(a,b):a,b \in A,b{\text{ }}is{\text{ }}exactly{\text{ }}divisible{\text{ }}by{\text{ }}a} \right\}$ so we have find the elements pair such that $(a,b):a,b \in A,b{\text{ }}is{\text{ }}exactly{\text{ }}divisible{\text{ }}by{\text{ }}a$. After this write all elements in $\left\{ {} \right\}$ to represent set R in roster form. Then the domain of R is given by all sets of the values which go into a function so writing all values of “a” in $(a,b)$ of roster form of set R in $\left\{ {} \right\}$. And range of R is given by all sets of the values which come out of a function so writing all values of “b” in $(a,b)$ of the roster form of set R in $\left\{ {} \right\}$.

Complete step-by-step solution:
Here the given set is \[A = \left\{ {1,2,3,4,6} \right\}\].
The set R is defined by $\left\{ {(a,b):a,b \in A,b{\text{ }}is{\text{ }}exactly{\text{ }}divisible{\text{ }}by{\text{ }}a} \right\}$. So let’s find first all elements of set R. Here the given condition is that in set $(a,b)$, b is exactly divisible by a and also $a,b \in A$.
So, possible values of a and b is 1,2,3,4,6. Firstly making the elements for set R.
Number 1 is exactly divisible by 1.
Number 2 is exactly divisible by 1 and 2.
Number 3 is exactly divisible by 1 and 3.
Number 4 is exactly divisible by 1,2 and 4.
Number 6 is exactly divisible by 1,2,3 and 6.
So, elements of set R are $(1,1),(2,1),(2,2),(3,1),(3,3),(4,1),(4,2),(4,4),(6,1),(6,2),(6,3),(6,6)$
Writing all elements of set R in $\left\{ {} \right\}$ will represent the roster form of set R,.
(A) So roster form of set R is given by $R = \left\{ {(1,1),(2,1),(2,2),(3,1),(3,3),(4,1),(4,2),(4,4),(6,1),(6,2),(6,3),(6,6)} \right\}$
(B) The domain of set R is given by all sets of the values which go into a function so writing all values of “a” in $(a,b)$ of roster form of set R in $\left\{ {} \right\}$.
So, here the input values are 1,2,3,4,6. So, Domain of set R is \[\left\{ {1,2,3,4,6} \right\}\]
(C) The range of set R is given by all sets of the values which come out of a function so writing all values of “b” in $(a,b)$ of roster form of set R in $\left\{ {} \right\}$.
So, here the output values are 1,2,3,4,6. So, the range of set R is \[\left\{ {1,2,3,4,6} \right\}\].

Note: For a given function the Codomain and Range are both on the output side, but are subtly different. The Codomain is the set of values that could possibly come out. The Codomain is actually part of the definition of the function. The range is a subset of the codomain. If the function is not fully known then we define the codomain instead of the range. In the above case we can define co-domain as \[\left\{ {1,2,3,4,6} \right\}\] because all elements of set A are coming in the range.