Given that U=(all the letters of the alphabet), X=(a,e,i,o,u), Y=(e,b,c,d,f,h). find (a). (X U Y)', (B) X' n Y' (C) X' n Y (D) X' U Y
To find the solutions to the given set operations, U=(all the letters of the alphabet), X=(a,e,i,o,u), and Y=(e,b,c,d,f,h), we will need to understand the basic principles of set theory.
(a) (X U Y)'
First, let's find the union of sets X and Y, denoted as (X U Y).
(X U Y) = {a, e, i, o, u, b, c, d, f, h}
The apostrophe (') represents the complement of a set. Therefore, (X U Y)' refers to the complement of (X U Y).
Since U represents all the letters of the alphabet, (X U Y)' = U - (X U Y).
(U - (X U Y)) = {b, c, d, f, g, j, k, l, m, n, p, q, r, s, t, v, w, x, y, z}
So, the answer to (a) is {b, c, d, f, g, j, k, l, m, n, p, q, r, s, t, v, w, x, y, z}.
(b) X' n Y'
To find the intersection of X' (complement of X) and Y' (complement of Y), we need to find the elements that are common to both sets.
X' = U - X = {b, c, d, f, g, h, j, k, l, m, n, p, q, r, s, t, v, w, x, y, z}
Y' = U - Y = {a, d, e, g, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z}
The intersection of X' and Y' (X' n Y') = {d, j, k, l, m, n, p, q, r, s, t, v, w, x, y, z}
So, the answer to (b) is {d, j, k, l, m, n, p, q, r, s, t, v, w, x, y, z}.
(c) X' n Y
To find the intersection of X' and Y, we need to find the elements that are common to both sets.
X' = U - X = {b, c, d, f, g, h, j, k, l, m, n, p, q, r, s, t, v, w, x, y, z}
Y = {e, b, c, d, f, h}
The intersection of X' and Y (X' n Y) = {b, c, d, f}
So, the answer to (c) is {b, c, d, f}.
(d) X' U Y
To find the union of X' and Y, we need to find all the elements in both sets without any duplication.
X' = U - X = {b, c, d, f, g, h, j, k, l, m, n, p, q, r, s, t, v, w, x, y, z}
Y = {e, b, c, d, f, h}
The union of X' and Y (X' U Y) = {b, c, d, e, f, g, h, j, k, l, m, n, p, q, r, s, t, v, w, x, y, z}
So, the answer to (d) is {b, c, d, e, f, g, h, j, k, l, m, n, p, q, r, s, t, v, w, x, y, z}.