Question

math 104
exam 1
in order to receive credit you will have to provide adequate explanation as to how you arrived at an answer. answers given with no supporting work will not receive credit. answers that are correct but are accompanied by incorrect steps that would imply a different answer than that given will also not be awarded credit.
some rules

• during the exam you must keep your arms and hands on or above the desk at all times. failure to do this will result in a warning, a second occurrence will result in a grade of zero for the exam.
• you must leave all paper on the desk at all times while completing the exam. you may only pick up your paper when you are prepared to submit the exam.
• the only electronic device you may have out during the exam is a calculator. if another device is seen during the exam this will result in an automatic grade of zero.
• if you have questions during the exam raise your hand and i will come over.

1. let (a = {1,2,3,5,a,c,f}), (b={2,3,4,b,e,f,w}), (c = {5,a,c,w}), and let (u={1,2,3,4,5,6,7,a,b,c,d,e,f,w,x,}) be the universal set for the problem.

(a) (10) draw a venn - diagram that includes circles for the sets a, b, and c. place all of the elements of u into the correct nonoverlapping regions of the diagram.
(b) (8) determine the following.
i. (acap c)
ii. (bcup c)
iii. (acap b)
iv. (acup c)

1. suppose that u is a universal set and a and b are subsets of u such that,

i. (n(acap b) = 45)
ii. (n(b)=200)
iii. (n(a)=120)
iv. (n(acup b)=200).
(a) (10) draw a venn - diagram with two circles for the sets a and b, and label each of the non - overlapping regions with the correct number of elements.
(b) (8) determine the following.
i. (n(u))
ii. (n(acup b))
iii. (n(acup b))
iv. (n((acap b)cup(acap b)))

Explanation:

Step1: Recall set - intersection definition

The intersection of two sets \(A\) and \(C\), denoted as \(A\cap C\), is the set of all elements that are in both \(A\) and \(C\). Given \(A = \{1,2,3,5,a,c,f\}\) and \(C=\{5,a,c,w\}\), we find the common elements.
\[A\cap C=\{5,a,c\}\]

Step2: Recall set - union definition

The union of two sets \(B\) and \(C\), denoted as \(B\cup C\), is the set of all elements that are in \(B\) or \(C\) (or both). Given \(B = \{2,3,4,b,e,f,w\}\) and \(C=\{5,a,c,w\}\), we combine the elements without repetition.
\[B\cup C=\{2,3,4,b,e,f,w,5,a,c\}\]

Step3: First find the complement of \(A\)

The complement of set \(A\), \(A'\), is the set of all elements in the universal set \(U\) that are not in \(A\). Given \(U=\{1,2,3,4,5,6,7,a,b,c,d,e,f,w,x\}\) and \(A = \{1,2,3,5,a,c,f\}\), then \(A'=\{4,6,7,b,d,e,w,x\}\). Then, find the intersection of \(A'\) and \(B\).
\[A'\cap B=\{4,b,e,w\}\]

Step4: First find the complements of \(A\) and \(C\)

We know \(A'=\{4,6,7,b,d,e,w,x\}\) and for \(C=\{5,a,c,w\}\), \(C'=\{1,2,3,4,6,7,b,d,e,f,x\}\). Then find the union of \(A'\) and \(C'\).
\[A'\cup C'=\{1,2,3,4,6,7,b,d,e,f,w,x\}\]

Answer:

i. \(A\cap C = \{5,a,c\}\)
ii. \(B\cup C=\{2,3,4,b,e,f,w,5,a,c\}\)
iii. \(A'\cap B=\{4,b,e,w\}\)
iv. \(A'\cup C'=\{1,2,3,4,6,7,b,d,e,f,w,x\}\)