21 If \( \mathrm{X}=\{6,4,2,0,-2,-4,-6\} \) and R is a relation on X where " a R b" means " a is the additive inverse of b " for each \( \mathrm{a} \in \mathrm{X}, \mathrm{b} \in \mathrm{X} \) Write R and represent it by an arrow diagram and show with reason if R is a function or not, and if R is a function , mention its range.

To define the relation \( R \) on the set \( X = \{6, 4, 2, 0, -2, -4, -6\} \) where "a R b" means "a is the additive inverse of b", we first need to identify the pairs \( (a, b) \) such that \( a + b = 0 \). ### Step 1: Identify the pairs in R The additive inverse of a number \( a \) is defined as the number \( b \) such that \( a + b = 0 \). Therefore, for each element \( a \) in \( X \), we can find its additive inverse \( b \): - For \( a = 6 \), the additive inverse is \( b = -6 \) (since \( 6 + (-6) = 0 \)). - For \( a = 4 \), the additive inverse is \( b = -4 \) (since \( 4 + (-4) = 0 \)). - For \( a = 2 \), the additive inverse is \( b = -2 \) (since \( 2 + (-2) = 0 \)). - For \( a = 0 \), the additive inverse is \( b = 0 \) (since \( 0 + 0 = 0 \)). - For \( a = -2 \), the additive inverse is \( b = 2 \) (since \( -2 + 2 = 0 \)). - For \( a = -4 \), the additive inverse is \( b = 4 \) (since \( -4 + 4 = 0 \)). - For \( a = -6 \), the additive inverse is \( b = 6 \) (since \( -6 + 6 = 0 \)). ### Step 2: Write the relation R Now we can write the relation \( R \) as a set of ordered pairs: \[ R = \{(6, -6), (4, -4), (2, -2), (0, 0), (-2, 2), (-4, 4), (-6, 6)\} \] ### Step 3: Represent R by an arrow diagram An arrow diagram can be represented as follows: ``` 6 → -6 4 → -4 2 → -2 0 ↔ 0 -2 → 2 -4 → 4 -6 → 6 ``` ### Step 4: Determine if R is a function A relation is a function if every element in the domain (the first component of each ordered pair) is associated with exactly one element in the codomain (the second component of each ordered pair). In this case, each element in \( X \) has exactly one corresponding additive inverse in \( X \). Therefore, \( R \) is indeed a function. ### Step 5: Determine the range of R The range of a function is the set of all second components (outputs) of the ordered pairs in the relation. From the pairs in \( R \): - The outputs are: \( -6, -4, -2, 0, 2, 4, 6 \). Thus, the range of \( R \) is: \[ \text{Range}(R) = \{-6, -4, -2, 0, 2, 4, 6\} \] ### Conclusion - The relation \( R \) is a function. - The range of \( R \) is \( \{-6, -4, -2, 0, 2, 4, 6\} \).