00:00 Good day, my name is Mr. Afeez, I will be your furthermaths teacher, and the topic is Logic

00:19 Reasoning.

00:20 At the end of this topic, you should be able to understand logic, caste it means, negation,

00:29 conditional statements and by conditional statements, identify the antecedents and consequence

00:37 of the simple statements.

00:41 Now let's define logical statements.

00:44 Illogical statements is the declaration of the verb or routine that is either true or

00:49 false, but not both.

00:51 It means any statement that is logical.

00:56 It is either true or false, but cannot be true and as well a false.

01:04 That is what it means.

01:05 So your statements can either be true or false, but cannot be true and still be false.

01:12 A true statement as a true table, as a true value T, that is if a statement is true,

01:19 the value is going to be T.

01:21 And if a statement is false, the value is going to be F. And logical statements are denoted

01:28 by letters PQ up.

01:30 That is when you want to denote analytical statements, you use a small letter PQ up.

01:39 Notes, questions, commands, expressions, exclamation, feelings, and not logical statements, meaning

01:49 that when you command someone is not a statement, when you exclamation is not a statement, when

01:57 you command is not a statement, when you express a feeling is not a statement, it is not all

02:03 these are what they are not statements.

02:06 Now let's look at this example one to know the difference between those statements and

02:13 those are not statements, which are the following are logical statements.

02:18 Number one, Nigeria is an Africa country, this is the statements, you know we said the

02:23 statement would be a verb or a routine expression, who is E, this is a question.

02:30 So it is not a statement, if I wrong, I shall not be late, this is the statement.

02:38 Japanese are adroking people, this is the statement, what a lovely man, this is an exclamation,

02:45 so it is not a statement, go and eat your food, this is a command, it is not a statement.

02:51 So, Nigeria is an African country, it is a statement and the rest, so we should not

02:56 defend between those that are not statements and those that are statements, negation statements.

03:07 You know we said earlier that if you want to denote any statement, we use a small letter.

03:14 So given a statement P, if a statement is P for example, the negation of P of that statement

03:21 is written as negates P, this sign it means not P, this not P is this statement, it means

03:33 it is false that P, it is false that P, for example, we have, Nigeria is a country, we want

03:44 to negate the statement, you said it is false that Nigeria is a country and if you want

03:52 to use, if you don't want to use, it is false, you can say Nigeria is not a country.

03:58 So is that how you said you want to negate any statement, is that how you say Nigeria is

04:02 a what is not a country or you said it is false that Nigeria is a country.

04:12 Now if P is true, that is if the statement P is true, the negation of that is going to

04:19 be what is going to be false and we know we said a true statement or a logical statement

04:25 is going to be noted by C and the false statement is going to be what if and if P is false,

04:31 then we have, the not P is going to be what is going to be true.

04:38 From this table you can see, we have a statement P and the statement is true.

04:44 The negation of that statement is going to be false.

04:49 You have a false statement, the statement is false and the negation of that statement is going

04:55 to be what is going to be true.

04:57 So if you have a statement which is true, the negation is going to be false and if you have

05:03 a statement which is false, the negation of that statement is going to be true.

05:11 Number two, let P be the statement, the negation is a rich country and then the negation of

05:20 this is the statement, it is false that the negation of that statement is a rich country

05:24 or simply say the negation of that is not a rich country.

05:32 Example number three, we can also have a number to be our illogical statement.

05:39 Let R be a statement, 3 plus A, 3 plus 4 equals 8.

05:45 Then the negation of that statement of 3 plus 4 equals 8.

05:49 The negation is going to be 3 plus 4 is not equals 8.

05:57 Now another thing that we should understand is what we call the conditional statements.

06:02 What is the conditional statement?

06:04 Let's kill stand for statement, family, family is the brilliant students and let's up stands

06:12 for the statement, family, past the test.

06:16 These are two simple statements and we want to explain what the conditional statement from

06:22 these two statements.

06:24 One way of combining the two statements, that is, the statement Q and statements, up is

06:31 by user we call conditional statements and that is, if family is the brilliant students,

06:38 then family past the test and we can do that in form of if Q then how.

06:47 If you have Q to be our statement, family is the brilliant students and how to be a statement,

06:53 family past the test.

06:54 So we want to do that in a conditional statement, you said if family is the brilliant students,

07:01 then family past the test or simply say if Q then how.

07:06 The students, if Q then how is combination of two simple statements, Q and how.

07:12 Many of that, the two statements that we have, P and Q, they are simple statements.

07:17 It is called a compost statement.

07:19 So whenever we have a two simple statements, we call it a compound statement and how can

07:25 we symbolize, we can symbolically represent this a compound statement.

07:32 The compound statement can be written as follows.

07:37 If Q then how.

07:38 If Q is the first statement, the first simple statement and how is a second statement.

07:47 If you want to represent this in a conditional statement, you simply say if Q then how.

07:52 It means ask Q implies how.

07:57 If Q then how.

08:00 And if you want to write it in a symbolically way, we write, we say what Q implies how.

08:07 So the statement Q implies how is read, is read ask Q implies how.

08:17 What if Q then how or Q if how.

08:21 These are the way in which you can write a conditional statement.

08:25 We understand it.

08:26 And this symbol is an operation to separate a two simple statements or a compost statements.

08:33 In the compost statement, Q implies how.

08:37 The statement Q is called antecedents.

08:40 When you have the first statement, for example, like we said earlier, family is the

08:46 first statement, is the brilliant student is an antecedent is the first statement is the

08:51 first input statements, then family pass the test is what we call consequence statement.

08:59 You call what it's called consequence of Q.

09:02 So if you have to compensate, if you have two simple statements, the first statement is

09:07 going to be an antecedent and what second statement is going to be consequence.

09:12 So hard can represent the conditional statement in a true or for city table.

09:19 We can represent this as shown below.

09:22 If we have statements, two statements, Q and how, if the first statement is true, just like

09:28 family is the brilliant students, which is the first statements.

09:32 And second statement is family pass exam, these are, so these two statements are what they

09:38 are true.

09:40 But if Q is true and how is true, the conditional statement of these two statements is going to

09:47 be true.

09:48 If family is a brilliant student, then family pass the exam.

09:55 This is true.

09:57 Now if Q is true, that is if family passed, if family is a brilliant student, and then

10:06 family did not pass the exam, the statement is going to be false.

10:12 That the conditional statement is going to be a false statement.

10:17 If the statements, the antecedent is false and the consequence is also true, then we have

10:25 the conditional statement to be a true statement.

10:29 And if the antecedent is false and the consequence also is what is false, then the statement is going

10:36 to be true.

10:37 For example, if family is not a brilliant student, then it did not pass the exam.

10:47 You can see that.

10:48 So it means it's very true.

10:49 If we, if it is not a brilliant student, then it cannot pass the exam.

10:53 We understand it at all.

10:54 So that is how to represent the conditional statement in a true or falsity or table.

11:01 Example number four.

11:02 Let's use this example to explain further what we mean by the conditional statement.

11:09 If Q is the statement, I mean male, and how does the statement, the son, the rice, let's

11:16 not consider the statement or in the conditional statement, A, if I mean male, then the son

11:23 will rise.

11:25 This is the conditional statement.

11:28 You can see that this is true.

11:29 This is true.

11:30 It means the conditional statement is going to be true.

11:33 If I mean, if I'm a male, then the son will not, will not rise.

11:38 This is true and this is false.

11:41 It means the statement is not what is a false.

11:44 The conditional statement is going to be what is going to be a false statement.

11:47 And if I'm not a male, then the son will rise.

11:50 This is not true; this is the false statement and this is true.

11:54 So the conditional statement is going to be a false statement.

12:00 If I'm not a male, then the son will not rise.

12:02 This is the false statement.

12:03 Why this is also his false statement?

12:05 So the conditional statement is going to be a false and it's going to be a true statement.

12:09 So the statements, all the statements are both, they statement A, C and D are all true.

12:18 But B is no, it's not true.

12:21 We understand it at all.

12:23 Less also understand what we mean by a converse statement.

12:28 A converse statement is also part of illogical statement.

12:33 And how do we explain a converse statement?

12:37 Less the statement Q implies P. This is a conditional statement.

12:43 Don't forget we said this is what a conditional statement.

12:46 This is antecedent Y, this is, or Y, this is a consequence statement.

12:54 The statement Q implies P is called the converse of P implies Q.

13:00 If for example, you have P implies Q.

13:10 The converse of this statement is going to be Q implies P.

13:15 That is the converse of this statement.

13:17 It does not mean that this statement is not true whether it's false.

13:21 The converse of this statement is going to be Q implies P.

13:26 EG, let's be the statement E triangle is equal angular.

13:31 And Q, the statement E triangle is equal at error.

13:37 The statement P implies Q means if a triangle is equal angular,

13:48 then it is equal at error.

13:50 You can see that this is a conditional statement.

13:53 And the converse of that conditional statement is going to be

13:56 if a triangle is equal at error, then it is equal angular.

14:02 We understand it at all.

14:03 So that is a converse statement.

14:05 When you have a conditional statement, the converse of that statement

14:08 is going to be the reverse.

14:10 That is, if you have P implies Q, then the converse of that is going to be Q implies P.

14:16 Now, let's look at the inverse statement.

14:20 The inverse statement, let's assume that the statement P, the statement P,

14:32 is implies Q is a conditional statement.

14:37 The inverse of this statement is going to be negation of P implies negation of Q.

14:47 Don't forget, we said this is negation of P.

14:50 That is the inverse of this statement.

14:52 And how do we identify an inverse statement?

14:56 If P is the statement, a triangle is equal angular, and Q is the statement,

15:03 a triangle is equal at error.

15:06 The inverse of this statement is, if triangle is not equal angular,

15:13 then it is not equal at error.

15:17 That is a converse, and we denote that to be not P implies not Q.

15:23 That is a converse statement.

15:24 Now, let's look at what we call a contra-postive statement.

15:29 What is a contra-postive statement?

15:31 If a statement, if you have a statement P, a conditional statement P, and Q,

15:40 the contra-postive statement of that is going to be negation of a converse statement.

15:45 You know, we said earlier, we said if P implies Q,

15:51 the converse of this statement is what is Q implies P.

15:57 Now, the contra-postive of this conditional statement is going to be the negation of a converse statement.

16:05 We understand this at all.

16:05 So that's what we call a contra-postive statement, the negation of a converse statement.

16:10 So we are going to negate this, we are going to negate also this.

16:14 And this is what a converse, the negation of that to give us a contra-postive statement.

16:21 It means, let's, for example, we have, I can swim.

16:25 And the statement Q is, we win.

16:29 The converse of this statement or the contra-postive of this statement is going to be,

16:34 if I cannot swim.

16:36 That is, are we, then I cannot swim.

16:41 We understand this at all.

16:42 So that is what we call a contra-postive statement.

16:46 I mean, the negation of a contra-of a converse statement.

16:50 Now, how do we identify, no, we explain what we call antecedents and consequences earlier?

16:57 Then how can we identify antecedents and a consequence in the conditional statement?

17:06 And that's also, we take also what we call a bi-conditional statement.

17:10 And what is a bi-conditional statement?

17:12 If P and Q are statements such that P implies Q, that is P,

17:19 that is, is conditional.

17:21 And the converse of that conditional statement,

17:26 if both conditional statement and the converse of that conditional statement are valid,

17:30 then P and Q imply each or each other.

17:34 Or P is equivalent to Q.

17:37 And we write that as what?

17:38 P imply Q.

17:41 There's a difference between this and this, this.

17:47 This is for a conditional statement.

17:52 But this is when you have a bi-conditional statement.

17:56 If you have a conditional statement and you also have a converse statement,

18:02 you have a bi-conditional statement and you have a converse statement.

18:06 If both conditional statements and converse of that converse statement are valid,

18:14 then we have what we call bi-conditional statements.

18:18 It means the statements P imply Q is called a bi-conditional statement of P and Q.

18:26 And the statements P and Q are equivalent to each other.

18:29 Do you understand that or so?

18:33 P imply Q could be read as Q is equivalent to P.

18:39 Or Q if and only if P or P if and only if Q.

18:48 Or if P then Q.

18:50 And if Q then P.

18:52 These are the ways in which we can represent our bi-conditional statement.

18:58 Let's now represent this bi-conditional statement in the true or falsity table.

19:05 And don't forget that this is the sign for bi-conditional statements.

19:11 That is when the two statements are equivalent.

19:15 Now, when a statement P is true and statement Q is also true,

19:22 the bi-conditional statement of that statement of that bi-conditional statement

19:28 or the equivalent of that two statements are going to be true.

19:33 That is if T is true, I mean if P is true and Q is also false,

19:39 then the bi-conditional statement is going to be a false statement.

19:43 If P is a false statement and Q is a true statement,

19:50 then the bi-conditional statement is going to be false.

19:55 Don't forget we say that it is only when the P and Q,

19:59 the conditional statements and the converse of that statement are valid,

20:03 then we can have what we call a bi-conditional statement.

20:06 So if for example now a bi-conditional statement is true when two substatements

20:12 are the same true value.

20:13 That is when we have a bi-conditional statement.

20:17 Whenever you have two substatements and what are those two substatements,

20:24 the antecedents and the consequence statements.

20:28 So when both of them are true, then we have what we call a bi-conditional true value, EG.

20:36 If P is the statement, the interior angles of the polygon are equal.

20:41 We have a P statement to be this and we have a key statement to be a polygon is regular.

20:50 Now how do we represent this in a bi-conditional statement?

20:54 It means P implies Q, it is statement, that is a bi-conditional statement.

21:00 If the interior angles of the polygon are equal, then the polygon is regular.

21:06 That is a what that is a bi-conditional statement.

21:12 Now if we want to, if we want, so because we know we said

21:17 for us to have equivalent statement, that is a bi-conditional statement,

21:21 we must have a bi-conditional statement being true and the converse of that is going to be also true.

21:28 So we have a bi-conditional statement,

21:30 BP implies Q and we have a converse of that bi-conditional statement,

21:40 which is if a polygon is regular, the converse of that,

21:43 then the interior angles of the polygon are equal.

21:49 It means P implies Q and Q implies Q, then you have what we call,

21:58 you have what we call, we have what we call an equivalent statement or a bi-conditional statement.

22:04 So P and Q have equivalence to each other, meaning that P is equivalent to Q.

22:09 And that's what we call a bi-conditional statement.

22:12 Now let's look at this example to understand what we mean by a bi-conditional statement.

22:18 Let's P be statements, Mary is a model, and let's Q be the statements, Mary is beautiful.

22:25 Let's not consider the statements, A, Mary is a model, if and only if.

22:31 Don't forget that we said there are ways in which we can identify or we can represent our

22:36 background statement. Mary is a model, if and only if she's beautiful. You can see that.

22:43 Mary can only be, can be a model, if and only if, if what if she's beautiful.

22:49 Mary is a model, if and only if she's, she's what she's ugly. This is a false statement.

22:57 Mary is not a model, if and only if she's what she is beautiful. Mary is not a model,

23:05 if and only if she's what she's ugly. So it means that the statement A and D are the true,

23:12 are what are true. Why B and C are, why B and C are the soft statements have the same true value.

23:19 The statement B and D and B and C are false because the soft statements that is the true statement,

23:25 Mary is a model, she's beautiful, have different true values. We understand it at all.

23:33 So that is how to represent a background statement in a true or falseity table.

23:42 Now have understood what we mean by logical statements and in negation, conditional statement and

23:50 background statement and the notification of an antecedent and consequence. We have this

23:57 evolution. The first is that to identify which of these statements is logical. The head is

24:05 conical in shape. If I think of my family, take the pencil away and we have these four and five.

24:14 So you are to identify whether these statements is logical or whether it is statements or is not

24:20 a statement. And the second one is a right negation of each of the following statements. You know,

24:26 we understand what mean by negation now. So it is a very, it sees very hot in the tropics.

24:32 E is an ensemble man. How can we negate? How can you write the negation of each of these statements.

24:37 And we can also write a negation of this statement by avoiding notes. We have been taught

24:43 that how to do that. So you have to use this as an evolution. E was presenting in school yesterday.

24:51 His friend is younger than my brother, without using the notes. Thank you for watching.

24:58 My name is Afiz. Till next time.