Answer to Question #216254 in Discrete Mathematics for Ree

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

Since the relation $\{(1, 3), (b, 3), (1, 4), (b, 2), (c, 2)\}$ contains two different pairs (1,3) and (1,4) with the same first coordinate, we conclude that this relation is not functional. In the rest cases there are no pairs with the same first soordinate, and hence all these relations are functional.

Answer: 1

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 $A$ .

Answer: 4