In this video i’m going to go over a copy of the midterm preparation quiz to help you get prepared for the midterm exam The first question on the proctored midterm will be the question that you see here and it will also include formulas that you see on the screen. So let’s go ahead and work on the questions So question number 2: the set containing all the elements that are common to both set A and set B is called the blank of set A and B. So elements that are common to both A and B is called the intersection Determine if the set is well defined or not well defined. The set of astronauts who walked on the moon So if i gave that description to anybody, several people they would all give me the same list. So this is a well-defined set Determine whether the set is finite or infinite. The set of odd numbers greater than 7. This is going to be infinite because it continues to go on and on and on, the numbers are greater than 7 Express the set in roster form. The set of natural numbers, so we need whole numbers, between 19 and 187. So we’re going to start at 20 and we’re going to go up to 186. So if we look in our choices down here, the correct answer for this is going to be D Express the set in set builder notation So we see the set contains natural numbers between 9 and 16 inclusive. So let’s take a look at what we’ve got down here for the choices. So it looks like the first choice is correct. Let’s just check on the other ones Yes and they wouldn’t include both endpoints so the correct answer here is going to be A. Write a description of the set. So the set V is the set of what? It’s letters in the alphabet from g to i, not to i inclusive. g to k inclusive. So it’s going to be choice C Determine whether the sets are equal, equivalent, both or neither So the set on the left has two elements and the set on the right also has two elements so we know they’re at least equivalent. But they don’t have the same elements so therefore these sets are simply equivalent. They’re not equal. Decide if a given statement is true or false So on the left side we see a single element 6 and then we look at the set on the right, the mathematical expression on the right and it’s a set with the elements 1 5 and 6. So this is a false statement and it’s a false statement because why? 6 is an element of that set on the right hand side. So the correct answer here is A Determine the number of subsets of the set that contains the elements 1, 2, 3 up to 9. So we know the formula for the number of subsets is two to the n, where n is the number of elements in this set. So this is two to the ninth and two to the ninth if i do that real quickly on a calculator. Let’s pull up the calculator on the computer No it’s wrong one pulled up, the wrong package there. Let’s get that out of the way. Let’s get the calculator up. So we want to go 2 and we want to raise it to the ninth power and we have 512. So the correct answer here is going to be 512 Draw a Venn diagram that illustrates the situation described. Set A and set B are disjoint which means they have nothing in common. So in the Venn diagram the two sets do not overlap and that’s choice D U is a set, the universal set, is a set of cities in a country. A is a set of cities that have a science museum. B is a set of cities that have a zoo We want to describe B complement in words. So B is a set of cities with a zoo. So B complement is going to be what the set of cities that do not have a zoo and that’s going to be choice C Okay so now we have a Venn diagram and we want us to pick where a particular country would fall. We have three categories: whether they produce corn, rice or wheat. So let’s look and see where country x falls. Do we see it in the first grouping? No. I don’t see it in the second grouping and I don’t see it in the third grouping So if it’s not in any of the three sets it must go into region number VIII. So the correct answer there would be region number VIII. Question 14 says use the Venn diagram to list the elements in the set A union B, take the complement of that in roster form. Now for some of these problems I have what I call detailed solutions because it was a little too hard to just write it on the screen for you and I wanted to explain how i did the problem. So from the Venn diagram we see that the universal set is all the elements under consideration and that’s the set of the elements 1 through 14 and actually that’s set b you know our answer b rather up there So let’s find A union B and if you look on the diagram here If i change my colors, all right I don’t see my drawing tool right now But i can do it with the drawing tool I have here. So A, no I’m not going drawing. Let’s go back over here. I was drawing over here. I don’t know why i can’t draw over here. Let’s take a look and see

Oh here let’s do red. So if i go in my diagram here and i take all the elements first that are in A union B, I’m really just circling here and I see that it’s all those elements there and if you put them all in a list you come up with right down here: 3, 4, 6, 7, 9, 11, 12 and 13, and then you want to take the complement of that and that’s everything that’s in the universal set that’s not included in A union B and I can do that by looking at the universal set and then crossing out the elements that I have put into A union B. I can also do it on the picture. If i change my color back here. If you notice everything outside of A union B is going to be elements right here. So it becomes 1, 2, 5, 8 and 10, which I have written right down here So the correct answer in this case is choice A, and I think i have a detailed solution for the next one. Yeah i do. This is a Venn diagram problem. The Greens are moving and the real estate agent located 81 houses for sale in their price range. Of the houses for sale and there’s a list of information given here and then we want to answer some questions. So let’s take a look at how we work this. So we know that there’s two sets: set A is going to be the house has a basement and set B it’s going to have a garage and here’s a general form of my two set Venn diagram So we’re going to start with region II How many houses had both and that’s 33. So that number is going to go right here into region II Now we know set A is basement and we know there’s a total of 46 with a basement but we’ve already accounted for 33 of them because they also have a garage. So we’re going to do subtraction and we’re going to find out that 13 is going to go right here in region I. They have only a basement, and by the same rationale, same reasoning I’ve got 48 houses that have a three-car garage but 33 of those also have a basement. So therefore I’m going to subtract the 33 and I’m going to come up with 15 houses that only have a garage. So here’s my Venn diagram so far and then I need to know how many don’t have either one and if you remember back up here it told us there were 81 houses total. If i add these three numbers together I get 71. So i have a 10 back here for region number IV. So here’s my completed Venn diagram. So how many houses had a finished basement but not a three-car garage and remember set A is basement and B is garage. So had a finished basement but not a three-car garage so this is going to be 13 right here 13. How many had a three car garage but not a finished basement that’s going to be 15, and then finally how many had either and remember we just added those up the 13 plus the 33 plus the 15 and that is 71, and I think I’ve got one more over here on the detailed solutions. Another Venn diagram problem. Mrs Bollo’s second grade class of 30 students conducted a pet ownership survey and the results show that 8 have a cat, 15 have a dog and 5 have both. How many students had no dogs? So again two sets. I’m going to let set A be cats and set B be dogs. I’m going to create the same two set Venn diagram. So let’s see there are eight students, set A is with a cat and eight students in this set. To find the region, remember there are five that had both so that’s going to go in region II Eight had a cat so 8 minus 5 is going to give me a three for region I Let’s see 15 had a dog but I already put 5 into region II so that’s going to give me 10 into region, this should be region II here. Let me cross that out that should be region II, and then when i put those in. I’ve got 3, 5 and 10 that’s 15. I’m sorry 3 and 5 is 8, 18 and there were 30 students in the class so 12 students don’t have a cat or a dog. So now we can answer the question: How many students have no dogs? All right so remember we said that set A or set B rather is dog. So no dog is going to be 3 plus 12 or 15 students. So the correct answer here is going to be 15 All right and I think i have detailed solutions for the next couple. No I don’t Okay let’s go back over here All right, make sure we got our pen going again Okay so we did that one, we did that one and we did this one. Okay so we’re up to number 17. The process of reasoning to a general conclusion through the observation of specific cases is called what kind of reasoning and that’s inductive reasoning. Use inductive to predict the next three numbers in the sequence. So what’s the pattern that we see: 12, minus 12, 12, minus 12. So the next three numbers are going to be 12, minus 12 and 12. That was a pretty easy one. Now this one i did give you a detailed solution for because it’s a little bit more complicated. So what’s the pattern we have? One triangle here, four triangles here, nine triangles here. So how do we get from figure one to figure two? We added how many triangles? Three. How do

we get from figure two to figure three? We added five triangles. So we added 3 here and we added 5 here. So reasonably what would we add here? Seven so that would take us up to sixteen. The other way to look at this is: this is 1 squared, this is 2 squared, this is 3 squared. So there’s another way to look at the pattern and this is 4 squared So two ways to look at that pattern Sometimes it’s helpful to see more than one pattern in a problem. Write negation of the statement. She earns more than me. So remember when we write a negation we want to look at the truth value of the original statement and let’s say that this statement is true. Let’s just say that was a true statement. So when we negate it has to be false. So what’s going to be the negation of this? She does not earn more than me and that’s going to have a false truth value. Okay and we want to translate into words. So we have the symbolic form here. They gave us the simple statements and it’s got a negation in front which means it is not true that. So I can eliminate the first two. They don’t begin with it is not true that and we start with the I eat bananas or this is an octopus. So the correct answer is the third choice, and now they want me to write this in a symbolic form I get an a which is p, and the play is boring q Okay number 23. I have a detailed solution for this one because it takes a bit more work on this one. So for number 23 we want to translate this into symbolic form. The doctor did not prescribe medicine so not p, but is and, the patient recover. So there it is symbolically and you can see in all four choices that’s the correct representation of the sentence. Now we have to build the truth table, So down here I put in my truth values. My p’s and my q’s Column number 1 is the negation of p: false, false, true, true, and then column 2 is the q truth values and then i want the connective. I’m using is and so I have to have at least two truths, both true simple statements and the only time i get that is in case number 3 and this correct answer is B, and number 24 I have over here also. In this case we’re doing like one row of a truth table. So I’ve given you a compound symbolic statement and then I’ve given you the truth values for the simple statements So it told me that p is false and I’ve got a not p in this. So then i know if p is false, not p is true and we’re going to replace the simple statements with their truth values. So not p is true. They told me q was true, p is false and r is true. So I’ve just put these and replace them in place of the simple statement symbols, and over here I’ve got true and true which gives me true. In my parentheses here I have false and true which gives me false. Now i’m going to negate that false to get a true. A true and a true gives me a true, and then the second situation they’ve changed the truth values now, Now p is false so therefore not p is going to be true and we don’t need to check on the other one so we’re going to replace not p with a true, q with the false, p with a false and r with a false. So on the left side I’ve got true and false. So I get false. On the right side I’ve got false and false which gives me a false Negate the false to get a true. False or true gives me true. So the compound statement is true Another detailed solution here. Construct a truth table for this statement here. So right down here I’ve got my truth table I’ve got first column is going to be the not t here, and then a second column is q. Connective here is and. So I’m looking for two trues which only happens in case three. Then i brought my negation of t’s over here and i have a conditional now. The only time that a conditional is going to be false is when i have a true to a false which i see in the fourth case. So I’ve got true. true. true. false and let’s see It’s C. C is the correct answer for that one, and then 26 similar to what we did before but this time we’re working with a conditional statement and again you can follow my work here I’m going to let you look through that you come out with a false on that one Now this one says use De Morgan’s laws or a truth table to determine whether the two statements are equivalent and we’ve got p or q and then we’ve got negation parentheses negation p or negation q. So I’m going to work with this part here because it looks like this De Morgan’s law, and instead of having p, I have not p, and

instead of having q, i have not q. So that’s where iIm going to start and look at the pattern the p went to not p. So my not p goes to I’ve got it actually the negation here. It’s going to go to a negation it’s going to go to p and then the not q remember the q goes to not q, so a not q is going to go to a q, and therefore and also the and statement goes to an or statement. So here’s what i end up with and these two statements are equivalent All right so then let’s go back over and so now we’re up to 28 All right, write the contrapositive of the statement. So remember contrapositive they’re flipped and negated. If the triangle is not an equilateral then the three sides of the triangle are not equal. So this is going to become the first the antecedent but not negated So if the sides of the triangle are equal then the triangle is equilateral and this is a true statement. So the correct answer here is B All right and now we’re moving into the geometry. So fill in the blank. Two lines in the same plane that do not intersect are, two lines in the same plane that do not intersect are parallel lines Okay and it wants you to identify the figure and notice the blue part of the figure here so. This is what’s called an open line segment and the answer is given, no i’m sorry given right here, that’s not correct, it’s this one here Sorry open line segment K G Classify the angle as acute, right, straight, obtuse, or none of these and you notice that little box in the corner there and that tells us this is a right angle. Find the requested angle and we want the complement. So I’ve got a detailed solution for this one. Remember complementary angles, their measures add to 90 degrees. So I took 90 and minus 36 and i got 54 which is answer B What was that? Okay this is a another one I have a detailed solution for. So we’ve got a little picture here. We’ve got two angles that are complementary. So that means their measures add up to 90. So I know that x plus 11x plus 42 must equal 90 Combine terms on the left side. So I get 12x plus 42 is 90 Subtract 42 from both sides and there should be a 42 i’m sorry it should be just 48. 12x equals 48 and then divide by 4 or x and you get divide by 12 rather and you get x equals 4. So the measure of angle 2, this is 4 degrees and then two ways to find the measure of angle 1. You can go back to this the expression for it which i did here and i come up with 86 I also know they’re complementary so i could simply subtract 4 from 90 to get the measure of angle 1, and then we’ve got one here with the transversal and they’ve given me the measurement of angle 8 here as 53 and they want me to find the measure of all the other angles. So there’s a lot of good relationships in here. So that we know angles, this should say 1, one and eight right here are supplementary. So we can find the measure of angle 1 by subtracting 53 from 180 so this one’s 127 and then you can continue working through this. These are vertical angles 1 and 7 so this is also 127 2 and 8 are vertical so this is 53 and then again. Oh let me get the eraser Oh there it is eraser. Let me erase this It’s not too clear. By some of the other relationships this should be 53 Basically what happens is if you come down here and work through them angles. Well all the odd angles are 127 and all the even angles are 53 degrees All right let’s go back over here. So we did that, we did that one. Okay what kind of a triangle is this? Two ways we want to do it by sides and angles. So we’ve got two sides that are equal there and we’ve got all the angles so it’s acute and it has to be isosceles right here because it has two equal sides, and what kind of quadrilateral is this? This is a parallelogram. Sorry i was looking off and not paying attention . Let’s try that again parallelogram and now we want to do similar triangles. So let’s go ahead and go over here. So we’ve got similar triangles and we want to find two sides here. We want to find the side x on this one and side y. So the first thing i did was for side x, I set up my ratio. So x is to 2.8 as 6 is to 2.4. Cross multiply 2.4x equals 6 times 2.8 which is 16.8 Divide both sides by 2.4 and you get a measure on this of 7. And then for y, same thing. I’m going to set up my similar triangle. So y is to 9 as 2.4 is just 6. Cross multiply, divide and you’ll come up

with y equal to 3.6 and so the correct answer here is C. 37, Okay 38. A triangle drawn on a map has size lengths of 9, 11, and 15. and the shortest of the corresponding real life distances is 126 kilometers. Find the length of the real longest of the real life distances. So I’ve made a picture here. I want to emphasize these are not right triangles. It’s just easier in Word to put a right triangle figure in and so we want to set up the proportions again like we had before. So we want to find the length in real life over here So x is to 15, the longest side of the one on the map, as the shortest side is to nine and again by cross multiplying and simplifying you come up with 210 which is answer D, and then find the area. Area of a triangle: one half of the base times the height. Here’s the base. Here’s the height. If you put that into the formula, you come up with 111 centimeters and watch your units centimeters squared Okay let’s go back over here to number 40. This is a perimeter problem Perimeter: you simply add the lengths of the sides. So 7 plus 8 is 15 and then again 15. So it’s going to be 30 and then they want to know what the units are and remember the units are the same units as the original measurements so it’s going to be yards, and i have a detailed solution for 41 This is a Pythagorean theorem and we want to find the length of side a. So a squared plus 36 squared is 39 squared. Square your numbers. Subtract 1296 from both sides. We get a squared is 225. So side a must be 15 and it’s inches And then they wanted to, I’m sorry then they wanted you to find the perimeter. So the perimeter is the distance around So that’s what I found here. I took the sum of the sides and i get 90 inches and then finally they wanted me to find the area, which is one half of the base times the height and I came up with 270 inches squared, and then find the circumference. Remember circumference: 2 pi r. You’re going to have these formulas available in the first question remember. So 2 pi r. So the radius here is 9. So 18 pi and they said use 3.14 for pi and if I multiply that out by it’s 3.14 by 18, I get answer D, and then in this one they want me to find the perimeter of the quadrilateral. First the area and then the perimeter. So the area of this is base times the height. Now the problem with this is notice the units are different. So we have to have the same units. So in order to get from meters and if you look in the answer choices they want the answer in centimeters. So 5 meters is equivalent to 500 centimeters. So 500 times 19 gives me 9,500 centimeters squared and then the perimeter is simply the distance around the outside. So 500 plus 27 plus 500 plus 27 gives me 1054. Find the surface area of a sphere with a radius of 12 inches using the formula and there’s the formula right there So if you put your numbers into the formula and if I use 3.14 I get 1808.64 and if I use my pi key I get a little tiny bit higher, Dither answer would be marked correct, and then finally find the surface area of the figure and remember surface area You have two of each side. So there’s two of the this side and this side, two of the long sides and then two of the top and the bottom and if you go ahead and put the values in given in the figure, you come up with a surface area of 32 square feet and that should help you get ready for your proctored midterm