Math Problem Statement
R is a relation from the set A = { x, 6, d, j, a, z, 5, s, 4, c }
to the set B = { q, e, 7, 6, r, 3, 9, x, 2, c }, and;
R = { ( z, 3 ), ( c, r ), ( 5, 9 ), ( x, q ), ( a, r ), ( 4, 2 ), ( s, x ), ( j, 6 ), ( 6, e ), ( d, 7 ) }
Choose a single pair that you can change in R so that R (after the change) will become an onto function from A to B. If such a pair does not exist, then choose the option: "None"
Question 14Select one:
( z, 3 )
( c, r )
( 5, 9 )
( x, q )
( a, r )
( 4, 2 )
( s, x )
( j, 6 )
( 6, e )
( d, 7 )
None
Solution
To determine the correct choice, let’s analyze the relation and decide how to make an onto function. Recall that an onto function (also called a surjection) means every element in must be mapped to by at least one element from .
Step 1: Verify the mapping of elements
The current relation is:
Elements in currently mapped:
- : mapped by ,
- : mapped by and ,
- : mapped by ,
- : mapped by ,
- : mapped by ,
- : mapped by ,
- : mapped by ,
- : mapped by ,
- : mapped by .
Missing element:
The only element in not mapped by any element of is .
Step 2: Modify to make it onto
To make onto, we need to add a mapping from to the missing element . However, since must also remain a function (each element in maps to exactly one element in ), we need to replace an existing pair in such that is mapped while no other elements in are left unmapped.
Replace :
Currently, maps to , so this element already covers ( 7. Replace function.
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Math Problem Analysis
Mathematical Concepts
Set Theory
Relations
Functions
Onto Functions (Surjections)
Formulas
-
Theorems
Definition of an Onto Function
Suitable Grade Level
Grades 11-12
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