Question

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)}

Answer 1

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)}