Solution to Suppose U = {1, 2, 3, 4, 5, a, b, c} is a universal set … - Sikademy
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Archangel Macsika

Suppose U = {1, 2, 3, 4, 5, a, b, c} is a universal set with the subset A = {a, b, c, 1, 2, 3, 4}. Answer questions 1 and 2 by using the given sets U and A. Question 1 Which one of the following relations on A is NOT functional? 1. {(1, 3), (b, 3), (1, 4), (b, 2), (c, 2)} 2. {(a, c), (b, c), (c, b), (1, 3), (2, 3), (3, a)} 3. {(a, a), (c, c), (2, 2), (3, 3), (4, 4)} 4. {(a, c), (b, c), (1, 3), (3, 3)} Question 2 Which one of the following alternatives represents a surjective function from U to A? 1. {(1, 4), (2, b), (3, 3), (4, 3), (5, a), (a, c), (b, 1), (c, b)} 2. {(a, 1), (b, 2), (c, a), (1, 4), (2, b), (3, 3), (4, c)} 3. {(1, a), (2, c), (3, b), (4, 1), (a, c), (b, 2), (c, 3)} 4. {(1, a), (2, b), (3, 4), (4, 3), (5, c), (a, a), (b, 1), (c, 2)}

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Here's the Solution to this Question

Question 1

A function is a relation in which each input has only one output.

In the relation {(1, 3), (b, 3), (1, 4), (b, 2), (c, 2)} the input x = 1 has multiple outputs: y = 3 and y = 4,

the input x = b has multiple outputs: y = 3 and y = 2.


1. {(1, 3), (b, 3), (1, 4), (b, 2), (c, 2)}


Question 2

1. Since 

range \{(1, 4), (2, b), (3, 3), (4, 3), (5, a), (a, c), (b, 1), (c, b)\}




=\{4,b,3,a,c,1,b\}\ne A,

we conclude that this function is not surjective.


2. Since 

range \{(a, 1), (b, 2), (c, a), (1, 4), (2, b), (3, 3), (4, c)\}

 

=\{1,2,a,4,b,3,c\}= A,

but 

domain \{(a, 1), (b, 2), (c, a), (1, 4), (2, b), (3, 3), (4, c)\}

 

=\{a,b,c,1,2,3,4\}\ne U

we conclude that this is not a surjective function from U to A.


3. Since 


range \{(1, a), (2, c), (3, b), (4, 1), (a, c), (b, 2), (c, 3)\}

 

=\{a,c,b,1,2,3\}\ne A,

we conclude that this function is not surjective.


4. Since 

range \{(1, a), (2, b), (3, 4), (4, 3), (5, c), (a, a), (b, 1), (c, 2)\}




=\{a,b,4,3,c,a,1,2\}= A


and

domain \{(1, a), (2, b), (3, 4), (4, 3), (5, c), (a, a), (b, 1), (c, 2)\}

 

=\{1,2,3,4,5,a,b,c\}=U,

we conclude that this function is surjective function from U to A .


4. {(1, a), (2, b), (3, 4), (4, 3), (5, c), (a, a), (b, 1), (c, 2)}


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Question ID: mtid-5-stid-8-sqid-2847-qpid-1404