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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)}