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Let S = { 1 , 2 , 3 , 4 } .  Then the number of elements in the set  is……………….

{ f : S × S : f     i s     o n t o     a n d     f ( a , b ) = f ( b , a ) ≥ a ∀ ( a , b ) ∈ S × S }

0 2 Views | Posted 2 months ago
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    Answered by

    alok kumar singh | Contributor-Level 10

    2 months ago

      ( 1 ,     1 ) ( 1 ,     4 ) ( 4 ,   1 ) ( 2 ,     4 ) ( 4 ,     2 ) ( 3 ,     4 ) ( 4 ,     3 ) ( 4 ,     4 )  all have only one image.

    (2, 1) (1, 2), (2, 2) each element has 3 choice.

    (3, 2) (2, 3) (3, 1) (1, 3) (3, 3) each element has two choices.

    ∴ total function = 3 × 3 × 2 × 2 × 2 = 72

    Case I

    None of the pre image have 3 as image, total functions = 2 × 2 × 1 × 1 × 1 = 4

    Case II

    None of the pre images have 2 as image then number of function = 25 = 32

    Case III

    None of the pre image have either 3 or 2 as image

    Total function = 15 = 1

    ∴ Total number of onto function

    = 72 – 4 – 32 + 1 = 37

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Kindly go through the solution 

 

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