1 00:00:18,030 --> 00:00:22,386 Welcome to the last video. In the Boolean series. In this 2 00:00:22,386 --> 00:00:26,742 video we will look at some of the more complicated Boolean 3 00:00:26,742 --> 00:00:28,326 expressions and therethrough tables. 4 00:00:29,640 --> 00:00:33,618 Now let's make matters a little bit more complicated. 5 00:00:33,618 --> 00:00:35,386 By using more brackets. 6 00:00:36,850 --> 00:00:38,870 P&R 7 00:00:39,980 --> 00:00:44,220 or Q&R. 8 00:00:46,260 --> 00:00:52,734 If then. P or not Q first thing that we want to establish 9 00:00:52,734 --> 00:00:57,174 is to find out how many different inputs are there. 10 00:00:58,150 --> 00:01:04,186 BRQRPQ there are altogether three different 11 00:01:04,186 --> 00:01:07,204 letters, so P. 12 00:01:09,180 --> 00:01:09,840 Q. 13 00:01:11,490 --> 00:01:11,950 R. 14 00:01:15,470 --> 00:01:20,126 Again, three different letters. Two to three gives me 8, so I'm 15 00:01:20,126 --> 00:01:23,618 going to have eight different input combinations or eight 16 00:01:23,618 --> 00:01:28,662 different rows in the truth table. Half of it is 4, so I'm 17 00:01:28,662 --> 00:01:32,542 going to start with four truths and then four forces. 18 00:01:34,440 --> 00:01:38,895 Then half of four is 2, so I'm going to have two 2, two fours 19 00:01:38,895 --> 00:01:40,677 until I fill up the table. 20 00:01:43,260 --> 00:01:45,410 Then just use true force. 21 00:01:46,650 --> 00:01:48,954 The force to fill in the last column. 22 00:01:52,160 --> 00:01:57,908 Now let's look at the different brackets. I have got a square 23 00:01:57,908 --> 00:02:02,534 bracket here. Which is combining together this bracket using the 24 00:02:02,534 --> 00:02:07,190 event operation. So to be able to use the if then operation 25 00:02:07,190 --> 00:02:12,234 first I need to know what is the value inside the square bracket, 26 00:02:12,234 --> 00:02:16,114 but unfortunately inside the square bracket I have got two 27 00:02:16,114 --> 00:02:19,994 different brackets. So first I need to calculate the P&R. 28 00:02:21,760 --> 00:02:24,994 Then I need to calculate the Q&R. 29 00:02:26,160 --> 00:02:29,527 Then I will be able to calculate what are the values in the 30 00:02:29,527 --> 00:02:32,894 square bracket, not to save some space. I'm just going to put the 31 00:02:32,894 --> 00:02:34,189 square brackets sign into here. 32 00:02:36,050 --> 00:02:40,574 Now once I've got the square bracket, I need to combine the 33 00:02:40,574 --> 00:02:44,344 values of the square bracket with this bracket, but inside 34 00:02:44,344 --> 00:02:50,376 this bracket I have got an not Q as well. So first I will need to 35 00:02:50,376 --> 00:02:55,277 calculate the value of not Q, then I will be able to calculate 36 00:02:55,277 --> 00:03:01,309 the P or not Q values and once I've got that I will be able to 37 00:03:01,309 --> 00:03:05,456 use the value for the event operation. And because there are 38 00:03:05,456 --> 00:03:07,341 no other if their operations. 39 00:03:07,450 --> 00:03:10,474 I can use the even operation here instead of having to write 40 00:03:10,474 --> 00:03:13,498 the whole thing out because then that would again just take up 41 00:03:13,498 --> 00:03:14,758 too much space in here. 42 00:03:15,510 --> 00:03:19,746 So let's get started P&R I'm using these two columns, the P 43 00:03:19,746 --> 00:03:24,335 and they are columns and I'm using an end again. What do I 44 00:03:24,335 --> 00:03:29,277 know about the end and is only true if both of them are true? 45 00:03:29,277 --> 00:03:34,219 This is true for both of them. That is not so it will be 46 00:03:34,219 --> 00:03:38,455 forced. That's true for both of them. That's going to give me 47 00:03:38,455 --> 00:03:43,397 true that is not gives me a force that is not give me a 48 00:03:43,397 --> 00:03:45,162 force force force and force 49 00:03:45,162 --> 00:03:51,414 again. Then I'm going to have to use the Q&R so Q&R both of them 50 00:03:51,414 --> 00:03:55,806 are true, gives me back a true value. They are different. That 51 00:03:55,806 --> 00:03:58,002 gives me back of force value. 52 00:03:58,930 --> 00:04:01,911 They bought force that still gives me a force value because 53 00:04:01,911 --> 00:04:07,122 it's the end. Both through gives me true Force force force. Now 54 00:04:07,122 --> 00:04:13,392 that's great. What I need to do next is to find out what is the 55 00:04:13,392 --> 00:04:17,572 overall value combinations in the square bracket. Now in this 56 00:04:17,572 --> 00:04:20,916 work square bracket I'm combining together these two 57 00:04:20,916 --> 00:04:26,350 using the OR. So this is this column and that is this column 58 00:04:26,350 --> 00:04:31,784 and I'm going to need to combine them together using the OR. What 59 00:04:31,784 --> 00:04:33,874 do I know about the? 60 00:04:33,960 --> 00:04:39,420 Or the OR is only force if both of them are forced in every 61 00:04:39,420 --> 00:04:44,100 other case, is the OR returns true? So that's true. This is 62 00:04:44,100 --> 00:04:50,648 force. That is true. That is force true Force, force 63 00:04:50,648 --> 00:04:54,792 force. So I've got the square bracket. 64 00:04:56,200 --> 00:05:01,030 Next thing I need is the note QQ is here. 65 00:05:02,060 --> 00:05:06,332 So they're not. Q would be just where Q is true, not 66 00:05:06,332 --> 00:05:07,400 QB LB force. 67 00:05:08,500 --> 00:05:12,493 An when Q is force not not Q will be true. 68 00:05:19,200 --> 00:05:24,640 I've got the not key now and what I need is the P or not Q. 69 00:05:26,230 --> 00:05:28,617 P or not, QP is this one. 70 00:05:30,440 --> 00:05:35,718 And not Q is this one that they are quite far away from each 71 00:05:35,718 --> 00:05:39,865 other, but it just an or operation. Again, if at least 72 00:05:39,865 --> 00:05:44,766 one of them is true, then the orbital return are true. So true 73 00:05:44,766 --> 00:05:49,290 force gives me true. True force gives me true true. True gives 74 00:05:49,290 --> 00:05:54,568 me 222 gives me true force and force gives me a force force and 75 00:05:54,568 --> 00:05:59,846 force give me a force for so true gives me a true and false 76 00:05:59,846 --> 00:06:02,079 or true. Me too again. 77 00:06:03,270 --> 00:06:04,810 And then the last step. 78 00:06:05,540 --> 00:06:07,478 Is that I need to use? 79 00:06:08,100 --> 00:06:09,888 The If then operation. 80 00:06:10,630 --> 00:06:14,751 Which two columns do I need to use the Ethan operation for? I 81 00:06:14,751 --> 00:06:16,336 need to use for this? 82 00:06:18,400 --> 00:06:23,044 And I need to use for the square bracket. Now this is 83 00:06:23,044 --> 00:06:28,075 my a because this is what on the left side for the event 84 00:06:28,075 --> 00:06:32,719 symbol and this is my B because this is on the right 85 00:06:32,719 --> 00:06:35,041 side of the if then symbol. 86 00:06:36,260 --> 00:06:40,145 True if then true returns a true 87 00:06:40,145 --> 00:06:46,802 value. Force even true returns a true value true if then two 88 00:06:46,802 --> 00:06:52,324 returns at true Value Force if then true returns with true 89 00:06:52,324 --> 00:06:57,846 value. True if then force returns a force, why you force 90 00:06:57,846 --> 00:07:03,870 if then forced returns a true value force if then true returns 91 00:07:03,870 --> 00:07:09,392 a true value, and force Tipton true again returns 2 value. 92 00:07:10,650 --> 00:07:15,499 There is a curiosity about this to table because if you look at 93 00:07:15,499 --> 00:07:19,602 the final output values, there is only 1/4 output in here. 94 00:07:19,602 --> 00:07:21,094 Everything else is true. 95 00:07:23,440 --> 00:07:27,550 Let's finish this video with a very complicated truth value 96 00:07:27,550 --> 00:07:33,715 table. Let me show you what you have to do. If you have gold 4 97 00:07:33,715 --> 00:07:34,948 input values, so. 98 00:07:36,610 --> 00:07:39,748 PX or Q. 99 00:07:41,920 --> 00:07:43,990 And not R. 100 00:07:46,420 --> 00:07:48,268 If and only if. 101 00:07:49,640 --> 00:07:52,540 Not Q or S. 102 00:07:53,580 --> 00:08:00,062 Now I have got four different letters PQ R&S not 2 to 4 = 103 00:08:00,062 --> 00:08:05,618 16. So I'm going to have a rather long truth value table. 104 00:08:05,618 --> 00:08:09,785 So let's start building it up. I've got P. 105 00:08:11,810 --> 00:08:13,040 I've got Q. 106 00:08:14,740 --> 00:08:16,000 I've got our. 107 00:08:17,250 --> 00:08:19,378 And I've got ***. 108 00:08:21,680 --> 00:08:26,893 Now what's the half of 16 half of 16 is 8, so I'm 109 00:08:26,893 --> 00:08:30,903 going to have to start with putting down 8 truth. 110 00:08:35,250 --> 00:08:36,840 And eight forces. 111 00:08:42,770 --> 00:08:46,928 Just about then the next step I'm going to 112 00:08:46,928 --> 00:08:48,776 put down four truths. 113 00:08:50,890 --> 00:08:52,600 N 4 forces. 114 00:08:55,280 --> 00:08:56,840 4 two again. 115 00:08:58,800 --> 00:09:00,348 And for force again. 116 00:09:03,360 --> 00:09:08,959 Next, 1/2 of four is 2, so two through 2 fours. 117 00:09:10,710 --> 00:09:17,646 To to force force again until I finish with the whole table. 118 00:09:19,490 --> 00:09:25,483 And in the last one is just 3/4, alternating all the way down. 119 00:09:35,450 --> 00:09:40,478 OK, does this start up the table? Then what I need to 120 00:09:40,478 --> 00:09:44,668 look at is breaking up the brackets, finding out the 121 00:09:44,668 --> 00:09:48,858 priorities so there is a square bracket and within the 122 00:09:48,858 --> 00:09:53,886 square bracket I have got a normal bracket and are not so 123 00:09:53,886 --> 00:09:57,238 I can cut combine together PX or Q. 124 00:10:00,730 --> 00:10:03,870 Then I would need to find out the note R. 125 00:10:05,710 --> 00:10:09,230 In the five, being able to calculate that, I will be 126 00:10:09,230 --> 00:10:12,110 able to calculate what is inside the square bracket. 127 00:10:15,230 --> 00:10:19,104 Then once the square bracket has been done, then I need to look 128 00:10:19,104 --> 00:10:23,574 at what's on the other side of the if and only if symbol on the 129 00:10:23,574 --> 00:10:28,044 other side. I have got not Q or S to be able to calculate that. 130 00:10:28,044 --> 00:10:30,130 First I need to calculate not Q. 131 00:10:32,760 --> 00:10:35,955 Then I can do the not Q or S. 132 00:10:37,470 --> 00:10:42,995 An once that's been done, I can calculate the if and only if, 133 00:10:42,995 --> 00:10:48,095 which will give me the final output. So let's get started PX 134 00:10:48,095 --> 00:10:54,895 or Q. What do I know about the X or the X Rays, the xclusive, or 135 00:10:54,895 --> 00:11:00,420 which is only true if one or the other is exclusively true? So 136 00:11:00,420 --> 00:11:05,520 basically, if the inputs are different than the X, or will be 137 00:11:05,520 --> 00:11:07,220 true PX or Q? 138 00:11:07,850 --> 00:11:11,260 So the first 2 columns they are the same, so the 139 00:11:11,260 --> 00:11:12,500 actual returns are force. 140 00:11:16,230 --> 00:11:19,118 They are different. The EXO returns are true. 141 00:11:21,900 --> 00:11:25,950 They are different, so the exhaust air returns are true. 142 00:11:27,660 --> 00:11:32,882 And in here they are the same, so the X or returns are force. 143 00:11:35,230 --> 00:11:40,550 So the next column in R2 value table will be the not R. So 144 00:11:40,550 --> 00:11:45,490 where are is true, North Star will be force an when R is 145 00:11:45,490 --> 00:11:47,770 force, not R will be true. 146 00:11:49,510 --> 00:11:55,894 Force Force 3 Three Force Force 147 00:11:55,894 --> 00:12:02,865 3. Two force for street due again, so I've got my starting 148 00:12:02,865 --> 00:12:07,410 values again. I need to calculate what's inside the 149 00:12:07,410 --> 00:12:12,460 square bracket, so what's inside the square bracket I'm combining 150 00:12:12,460 --> 00:12:18,520 together the X or and the North R using the end operation. 151 00:12:19,540 --> 00:12:23,110 What do you remember about the end operation? It's only 152 00:12:23,110 --> 00:12:27,037 true if both of them are true, so let's came through. 153 00:12:28,990 --> 00:12:32,770 That is, both of them are true, so I'm going to place a true 154 00:12:32,770 --> 00:12:33,850 sign there, same here. 155 00:12:35,900 --> 00:12:36,590 Here. 156 00:12:37,610 --> 00:12:38,350 Here. 157 00:12:40,200 --> 00:12:44,750 And the rest of them are all different or double forces, so I 158 00:12:44,750 --> 00:12:48,600 can just fill in the last of the table with force. 159 00:13:01,600 --> 00:13:07,915 Then calculate not Q while Q is here and not Q will be just the 160 00:13:07,915 --> 00:13:12,546 opposite of what Cindy Q column so true will be force. 161 00:13:17,170 --> 00:13:19,318 Annefors will be true. 162 00:13:36,740 --> 00:13:42,031 And then what I need to do here is combined together, not Q. 163 00:13:43,870 --> 00:13:45,170 An S. 164 00:13:48,290 --> 00:13:52,008 Using the OR operation. So what do I remember about the 165 00:13:52,008 --> 00:13:56,402 OR the OR is only force if both of them are force? So 166 00:13:56,402 --> 00:13:59,444 let's try to filter out those inputs, not both. 167 00:13:59,444 --> 00:14:03,162 Force both of them are force, so this will be a 168 00:14:03,162 --> 00:14:04,176 force in here. 169 00:14:05,640 --> 00:14:08,430 Both of them are force, so that's another force. 170 00:14:12,620 --> 00:14:15,905 Both of them are force. Here is another force. 171 00:14:17,490 --> 00:14:18,810 Here is one more. 172 00:14:22,280 --> 00:14:25,459 So the rest of them I can fill in with truth. 173 00:14:35,990 --> 00:14:41,795 Lovely, so the final answer is the if and only if and what do I 174 00:14:41,795 --> 00:14:46,052 combine together to get the final answer? On one side I've 175 00:14:46,052 --> 00:14:51,083 got the square bracket which is here and the other side I have 176 00:14:51,083 --> 00:14:56,888 called the note Q or S which is here. So if and only if returns 177 00:14:56,888 --> 00:15:02,694 true. If the inputs are the same and it returns force if the 178 00:15:02,694 --> 00:15:03,960 inputs are different. 179 00:15:05,030 --> 00:15:08,904 To enforce difference, so the if and only if gives me force force 180 00:15:08,904 --> 00:15:13,076 and force the same. So the if and only if. It gives me true. 181 00:15:13,950 --> 00:15:20,152 Different force, same true, different force, different 182 00:15:20,152 --> 00:15:26,354 force, same again true, same different same. 183 00:15:27,610 --> 00:15:28,440 Same. 184 00:15:29,970 --> 00:15:33,510 Different gives me a force different gives me a force 185 00:15:33,510 --> 00:15:37,404 different, gives me a force, different, gives me a force and 186 00:15:37,404 --> 00:15:42,006 a different at the end. Again gives me or force. So this is 187 00:15:42,006 --> 00:15:45,546 the final answer for this long and rather complicated Boolean 188 00:15:45,546 --> 00:15:49,794 expression. And as you see there are quite a bit of different 189 00:15:49,794 --> 00:15:51,564 combinations of force and truth. 190 00:15:53,530 --> 00:15:58,093 I hope that by now you have been able to build a good 191 00:15:58,093 --> 00:16:00,901 understanding of the Boolean expression and they threw 192 00:16:00,901 --> 00:16:04,411 tables. I have a few last practice questions for you 193 00:16:04,411 --> 00:16:07,921 and the answers will follow, so these are the practice 194 00:16:07,921 --> 00:16:08,272 questions. 195 00:16:14,380 --> 00:16:15,990 And here are the answers.