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 RR and decide how to make RR an onto function. Recall that an onto function (also called a surjection) means every element in BB must be mapped to by at least one element from AA.


Step 1: Verify the mapping of elements

The current relation is: R={(z,3),(c,r),(5,9),(x,q),(a,r),(4,2),(s,x),(j,6),(6,e),(d,7)}.R = \{ (z, 3), (c, r), (5, 9), (x, q), (a, r), (4, 2), (s, x), (j, 6), (6, e), (d, 7) \}.

Elements in BB currently mapped:

  • 33: mapped by zz,
  • rr: mapped by cc and aa,
  • 99: mapped by 55,
  • qq: mapped by xx,
  • 22: mapped by 44,
  • xx: mapped by ss,
  • 66: mapped by jj,
  • ee: mapped by 66,
  • 77: mapped by dd.

Missing element:

The only element in BB not mapped by any element of AA is 77.


Step 2: Modify RR to make it onto

To make RR onto, we need to add a mapping from AA to the missing element 77. However, since RR must also remain a function (each element in AA maps to exactly one element in BB), we need to replace an existing pair in RR such that 77 is mapped while no other elements in BB are left unmapped.

Replace (d,7)(d, 7):

Currently, dd maps to 77, 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