Question

Let G = {−2, 0, 2} and H = {4, 6, 8} and define a relation V from G to H as follows: For every (x, y) is in G ✕ H, (x, y) is in V means that x − y 4 is an integer. (a) Is 2 V 6? Yes No Is −2 V 8? Yes No Is (0, 6) is in V? Yes No Is (2, 4) is in V? Yes No (b) Write V as a set of ordered pairs. (Enter your answer in roster notation.) (c) Write the domain set of V. (Enter your answer in roster notation.) Write the co-domain set of V. (Enter your answer in roster notation.)

Answer 1

Answer:

a) Yes, No, No No

b) V={(-2,6),(0,4),(0,8),(2,6)}.

c) Domain={-2,0,2}

d) Co-domain={4,6,8}

Step-by-step explanation:

If A and B are two sets then the cartesian product of A and B denoted by $A\times B$ is the set of all ordered pairs (a,b), where $a\in A$ and $b\in B$

$A\times B$={$(a,b)|a\in A \:\wedge b\in B$}

The given relation is

$V:G\to H$ for every (x, y) is in G ✕ H, (x, y) is in V means that $\frac{x-y}{4}$ is an integer

a) $2V6=(2,6)=\frac{2-6}{4} =-1$ Yes, since -1 is an integer

b) $-2V8=(-2,8)=\frac{-2-8}{4} =-2.5$ No, since -2.5 is not an integer

c) $0V6=(0,6)=\frac{0-6}{4} =-1.5$ No, since -1.5 is not an integer

$2V4=(2,4)=\frac{2-4}{4} =-0.5$ No, since -0.5 is not an integer

G ✕ H={(-2,4),(-2,6),(-2,8),(0,4),(0,6),(0,8),(2,4),(2,6),(2,8)}

From this set, we have some ordered pairs that satisfy the definition of V.

V={(-2,6),(0,4),(0,8),(2,6)}.....The difference of these ordered pairs are multiples of 4.

c) The domain of V is the set of the first coordinates of the ordered pairs of V

Domain={-2,0,2}

The co domain of v is the set of all the second coordinates of the ordered pairs of V

Co-domain={4,6,8}