Sure! Let's go through each part of the question step by step.
### Given:
Sets [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]:
[tex]\[ P = \{2, 4, 6\} \][/tex]
[tex]\[ Q = \{4, 6, 10\} \][/tex]
Relation between [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]:
[tex]\[ \text{relation} = \{(2, 4), (4, 6), (6, 6)\} \][/tex]
We need to determine the following:
1. The image of [tex]\( 4 \)[/tex].
2. The object(s) of [tex]\( 6 \)[/tex].
3. The range of the relation.
### (a) The Image of [tex]\( 4 \)[/tex]:
The image of an element in set [tex]\( P \)[/tex] under a relation is the element in set [tex]\( Q \)[/tex] that is paired with it.
From the given relation:
[tex]\[ (2, 4), (4, 6), (6, 6) \][/tex]
We see that [tex]\( 4 \)[/tex] is paired with [tex]\( 6 \)[/tex] in the ordered pair [tex]\( (4, 6) \)[/tex]. Hence, the image of [tex]\( 4 \)[/tex] is:
[tex]\[ 6 \][/tex]
### (b) The Object(s) of [tex]\( 6 \)[/tex]:
The objects of an element in set [tex]\( Q \)[/tex] under a relation are the elements in set [tex]\( P \)[/tex] that are paired with it.
From the given relation:
[tex]\[ (2, 4), (4, 6), (6, 6) \][/tex]
We look for pairs where the second element is [tex]\( 6 \)[/tex]. We find:
[tex]\[ (4, 6) \][/tex]
[tex]\[ (6, 6) \][/tex]
Hence, the objects of [tex]\( 6 \)[/tex] are:
[tex]\[ 4 \text{ and } 6 \][/tex]
### (c) The Range of the Relation:
The range of a relation is the set of all second elements (outputs) in the ordered pairs of the relation.
From the given relation:
[tex]\[ (2, 4), (4, 6), (6, 6) \][/tex]
The second elements are:
[tex]\[ 4, 6, 6 \][/tex]
To form the range, we list these elements without repetition. Therefore, the range of the relation is:
[tex]\[ \{4, 6\} \][/tex]
### Summary:
(a) The image of [tex]\( 4 \)[/tex] is [tex]\( 6 \)[/tex].
(b) The objects of [tex]\( 6 \)[/tex] are [tex]\( 4 \)[/tex] and [tex]\( 6 \)[/tex].
(c) The range of the relation is [tex]\( \{4, 6\} \)[/tex].
