<p>Set A contains a members and set B contains b members.
If C is the set of all members of both A and B and contains c members. How many members are in either A or B, but not C?</p>
<p>The answer to this question is a+b-2c. I don't understand how! So set C has members a, b, and c, right? So shouldn't it just be a+b-c? So confusing..</p>
<p>{a, b, c, d, d}
The average (arithmetic mean) of the above set of numbers is equal to the mode. Which of the following could be the numbers in order from smallest to largest?</p>
<p>The answer to this is a,b,d,c or basically any order in which d is in one of the middle 2 numbers. Again..I have no idea how it works. Any help on these 2 would be appreciated!</p>
<ol>
<li><p>You have to subtract c again in order to get rid of the elements in C. For example, |A ∪ B| = a+b-c by the Inclusion-Exclusion principle. But this includes elements in both A and B, so we have to subtract c, giving us a+b-2c.</p></li>
<li><p>Assuming a,b,c,d are distinct, (a+b+c+d+d)/5 = d → a+b+c+d+d = 5d → a+b+c = 3d. So the average of a,b,c is d. Therefore d must be somewhere in the middle.</p></li>
</ol>
<p>A specific example for the first one might help. </p>
<p>Example: Let A = {a, b, c} and let B = {b, c, d, e}. Then C = {b, c} (for your information, C is called the intersection of A and B). Let’s let D be the set consisiting of elements in A or B but not C (this set actually has a special name too - it’s called the symmetric difference of A and B). D = {a, d, e}.</p>
<p>Note that A has 3 elements, B has 4 elements, C has 2 elements, and D has 3 elements. Now 3 + 4 - 2 = 5. This is the number of elements that are in A or B (also called the union of A and B), but this is not the answer to the question. </p>
<p>The answer to the question is 3 + 4 - 2(2) = 3. You have to subtract off the number of elements in C twice!</p>
<p>Summary: If you add up the number of elements in A and B, then subtract the number of elements in the intersection, you get the number of elements in the union of A and B.</p>
<p>If you add up the number of elements in A and B, then subtract the number of elements in the intersection twice, you get the number of elements in the symmetric difference of A and B.</p>
<p>The example did help me out but based on the information, shouldn’t C = {a, b, c, d, e} because it contains the elements of both set A and B? Including the set D helped me understand the concept of the union of A and B and the symmetric difference of A and B so thanks a lot!</p>
<p>In general, when you use the word “and” you are talking about the intersection. For the set you are describing (the union) you would use the word “or.”</p>
<p>Regarding your second question, this is also another way to look at it:</p>
<pre><code>The mode of a set of elements is the most repeated element in that set. In this case, the mode is d. The question also states that the mode is equal to the average (arithmetic mean), so you have another piece of information: The average (arithmetic mean) is d.
Now, the trick is how to know the order of these numbers (elements) from least to greatest. Whenever you have a set of numbers, the mean is always in the middle. It can’t be at the edges.
For example: 3,5,4. To calculate the mean you would order it as 3,4,5. (3+4+5)/3= 4. So, your mean is 4. Did you notice that 4 was in the middle when you ordered the numbers?
Apply this to the question. Since the mean (which was also the mode) is d, so d MUST be somewhere in the middle.
</code></pre>