Question 202396
Given U = {All letters of the alphabet}, A = {c, d, e, f}, and B = {e, f, g, h, k}.<br>
To start with<ul><li>I am going to assume that U = {All the lowercase letters of the alphabet}. If this is incorrect, then the answers to (c), (d) and (e) will be incomplete.</li><li>To avoid any potential confusion, I will use U for the "universal" set and "u" for the union operator</li><li>When the elements of a set are listed, the order is not important</li><li>The union operator, u, is an inclusive operator. The union of two sets will <b>always</b> include every member of both sets. Elements of both sets, if any, are listed only once in the union. (Never list duplicate elements in any set.)</li><li>The intersection operator, &#8745;, is an exclusive operator. The intersection of two sets will include <b>only</b> elements that are members of <b>both</b> sets.</li></ul>
(a) A u B
This means "the set whose elements are members of set A or set B (or both)".
A u B = {c, d, e, f, g, h, k}<br>
(b) A &#8745; B
This means "the set whose elements are members of both set A and set B".
A &#8745; B = {e, f}<br>
(c) A&#8242; &#8745; B&#8242;
A&#8242; means "the set of all elements of the universal set, U, which are <b>not</b> members of set A.
A&#8242; = {a, b, g, h, i, j, k, l, m, ..., z}
B&#8242; means "the set of all elements of the universal set, U, which are <b>not</b> members of set B.
B&#8242; = {a, b, c, d, i, j, l, m, ..., z}
A&#8242; &#8745; B&#8242; means "the set whose elements are members of both set A' and set B'".
A&#8242; &#8745; B&#8242; = {a, b, i, j, l, m, ..., z}<br>
(d) A&#8242; u B&#8242;
This means "the set whose elements are members of set A&#8242; or set B&#8242; (or both)". Using A&#8242; and B&#8242; from (c) we get
A&#8242; u B&#8242; = {a, b, c, d, g, h, i, j, k, l, m, ..., z}<br>
(e) A u B&#8242;
This means "the set whose elements are members of set A or set B&#8242; (or both)". Using A and B&#8242; from (c) we get
A u B&#8242; = {a, b, c, d, e, f, i, j, l, m, ..., z}<br>
(f) (A u B&#8242;) &#8745; B
The first part of this is the answer to part (e). (A u B&#8242;) &#8745; B means "the set whose elements are members of the set B and members of the answer to part (e)"
(A u B&#8242;) &#8745; B = {e, f}<br>
(g) (A u B) &#8745; (A u B&#8242;)
The first part is the answer to part (a), The last part is the answer to part (e). So (A u B) &#8745; (A u B&#8242;) means "the set whose members are members of the answer to part (a) and members to the answer to part (e)". So
(A u B) &#8745; (A u B&#8242;) = {c, d, e, f} (which is set A}