Set Notation for Identifying Substances: Tests 1-3 and Homework Statement Help

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Dommm
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Homework Statement



Three tests are available to identify six different substances.

Test 1 is true in the presence of substance 1,2,3, and 4
Test 2 is true in the presence of substance 3,4, and 5
Test 3 is true in the presence of substance 2,5, and 6

Using set notation to denote the events that a substance under test is

a) substance 1
b) substance 2
c) substance 6

Homework Equations



Let [itex]E_{i}[/itex] denote the event that the i'th test is positive, [itex]i= 1, 2, 3[/itex]

The Attempt at a Solution



How do I properly represent the answers in set notation?

Currently here are my attempts at a,b, and c.

a) [itex]S_{1} = E_{1}[/itex] [itex]\cap[/itex] [itex]E_{2}^{'}[/itex] [itex]\cap[/itex][itex]E_{3}^{'}[/itex]

b) [itex]S_{2}[/itex] cannot be identified

c) [itex]S_{6} = E_{3}[/itex] [itex]\cap[/itex] [itex]E_{2}^{'}[/itex] [itex]\cap[/itex][itex]E_{1}^{'}[/itex]


Is this how you correctly write the answer in set notation?

Also with b) I am also unsure of my answer. Would it be correct to say that E1 is true AND not S6?
 
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Dommm said:

Homework Statement



Three tests are available to identify six different substances.

Test 1 is true in the presence of substance 1,2,3, and 4
Test 2 is true in the presence of substance 3,4, and 5
Test 3 is true in the presence of substance 2,5, and 6

Using set notation to denote the events that a substance under test is

a) substance 1
b) substance 2
c) substance 6

Homework Equations



Let [itex]E_{i}[/itex] denote the event that the i'th test is positive, [itex]i= 1, 2, 3[/itex]

The Attempt at a Solution



How do I properly represent the answers in set notation?

Currently here are my attempts at a,b, and c.

a) [itex]S_{1} = E_{1}[/itex] [itex]\cap[/itex] [itex]E_{2}^{'}[/itex] [itex]\cap[/itex][itex]E_{3}^{'}[/itex]

b) [itex]S_{2}[/itex] cannot be identified

c) [itex]S_{6} = E_{3}[/itex] [itex]\cap[/itex] [itex]E_{2}^{'}[/itex] [itex]\cap[/itex][itex]E_{1}^{'}[/itex]Is this how you correctly write the answer in set notation?

Also with b) I am also unsure of my answer. Would it be correct to say that E1 is true AND not S6?

You haven't said what ##S_i## means, but if ##S_i=\{i\}## then a) and c) look ok. For b) substance two is the only one that is true for tests 1 and 3, isn't it?
 
Last edited:
doh! I see. Thank you :)

How about the notation? Is the way that I represented the answers correct?
 
Dommm said:
doh! I see. Thank you :)

How about the notation? Is the way that I represented the answers correct?

I don't know what your class expects, but if you define the sets as you have then the relations you have written are correct.