hi to all (:
I just want to remind everyone that its our LOGIC TEST tomorrow!!
I'm kinda nervous about the test tomorrow because I have some difficulties understanding some topics.
I don't get these following topics:
=counterexample
=proof of contradiction (esp solving problems using graphs)
And that's everything:)..I hope everyone will do great on the test tomorrow!good luck!!
Showing posts with label Logic. Show all posts
Showing posts with label Logic. Show all posts
Tuesday, January 6, 2009
Review Class
Today we had a review class since Mr. K. gave us an extra day to prepare for the test. Mr. K. wasn't here today unforunately. For the class, we received a Logic Review sheet as seen below.
Some of the questons you will see on the sheet, we've already went over so it shouldn't be to hard to answer. It was also said that in the beginning of the class that the questions on the sheet were taken from past tests. So what you see on that sheet is basically a test in itself. So its safe to say, the questions seen are identical on what you will see on tomorrow's test.
Some of the questons you will see on the sheet, we've already went over so it shouldn't be to hard to answer. It was also said that in the beginning of the class that the questions on the sheet were taken from past tests. So what you see on that sheet is basically a test in itself. So its safe to say, the questions seen are identical on what you will see on tomorrow's test.As for question 9, you will need to make a chart to solve. I've made a chart for those of you who need it
As you may have noticed, there is a question mark in place of one of the points. I have yet to figure out what that value is. Most likely you will need to go trough the problem to be able to find the value.
Thats it for today's class.
Remember: Test Tomorrow
Good luck and study hard !
Next Scribe: Nohom
Monday, January 5, 2009
It's 2009! :D Scribe post for January 05, 2009
Happy New Year
&
Welcome Back To School!
Hii Everyone! ^-^
First of all, I hope that you all had a wonderful winter break :] That 2-week hibernation and fun was great, but now it's time to get up and have ourselves back on track. As what Mr.K said, "We only have 3 weeks, almost exact, before the exam. After Logic, we have one more unit to do." (Well, it was something like that :P) To warm up our brains, we had a Pre-test on Logic today. Our real test will be on Wednesday (January 07, 2008), since Mr.K said that we'd have one more day to review Logic a little more.
Here are the questions & answers/corrections if you've missed the Pre-test (They can also be seen in the slides):
(1) In a class of 50 students, 18 take music, 26 take art, and 2 take both art and music. How many students are not enrolled in either music or art? (a) 6 ; (b) 8 ; (c) 16 ; (d) 24
to explain it in the form of a Venn Diagram...
(2) What is true about the statement "If two angles are right angles, the angles have equal measure" and its converse "If two angles have equal measure then the two angles are right angles"?
(a) The statement is true but its converse is false.
(b) The statement is false but its converse is true.
(c) Both the statement and its converse are false.
(d) Both the statement and its converse are true.
(a) The statement is true but its converse is false.
(b) The statement is false but its converse is true.
(c) Both the statement and its converse are false.
(d) Both the statement and its converse are true.
- Converse is false because two angles that have equal measure doesn't necessarily have to be right angles. For example, an isoceles triangle has two equal angles but they are not right angles.
(3) The inverse of the converse of a conditional statement is the contrapositive.
(4) Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8}, B = {1,3,6,9}, and C = {1,3,4,6,8}. List the members in the set: ( A ∩ C )'
to explain in the form of a Venn Diagram...
%27.PNG)
(click to enlarge)
%27.PNG)
(click to enlarge)Therefore, the members in the set ( A ∩ C )' are 1, 2, 3, 5, 7, 9.
(5) Prove that the difference of square of two odd numbers is always divisible by 4.
=
=
=
=
This has a factor of 4 and is divisible by 4.
I think I explained them as well as I could. Tomorrow, we will have time to review Logic with the class once more. Homework for today was to try to prove if 
is rational or not. Look at the slide that was on December 19 for the explanation on 
because it is similar. Also, we have to finish other unanswered exercises that we were assigned including Exercise 50.
That is all for today! ^-^
The next scribe will be Randall :]
- CharChar
Friday, December 19, 2008
Proof by Contradiction
Hi everyone! Although this is the last day of school for this year, we still had lessons in class that we talked about for the whole period...
Before we went into groups and answer group questions, Mr. K. recalled the sentence that we talked about the other day:
This sentence is false.
This sentence is a Paradox which means it can't be true and false at the same time, but it should be true and false at the same time. A Paradox is one kind of Contradiction that goes round and round.
When we went into our groups, we were given logic questions that we needed to answer. Here's the first question.
On of Isle of View, a knight always tell the truth and a knave always lie. A person who just came met Mr. A and Mr. B. Mr. A said "one of us is a knave". Which one is the knave, and which one is the night?
We need to answer this question by Proof by Contradiction, which means we have to assume the opposite of the statement and prove that it's right. This was how we tackled this question.
First, we choose the Direct Reasoning of the statement, which in this case was
1) Assume Mr. A is a knight. We can then tell that
2) His statement is true, therefore our conclusion is that
3) Mr. B is a knave and Mr. A is a knight.
Using the Indirect Reasoning, we take the opposite statement and check if it leads to contradiction...
1) Assume Mr. A is a knave
2) His statement is false
3) They are both knights
This part of the reasoning leads to a contradiction, which was what we were looking for. Therefore, Mr. A is a knight.
We looked at two more questions that deals with logic, obviously :) It was pretty much straight forward... We do have homework, however that we need to do. It's Exercise 50 all questions.
Next scribe is Charizze
Pre-test on Monday - January 5, 2009
Before we went into groups and answer group questions, Mr. K. recalled the sentence that we talked about the other day:
This sentence is false.
This sentence is a Paradox which means it can't be true and false at the same time, but it should be true and false at the same time. A Paradox is one kind of Contradiction that goes round and round.
When we went into our groups, we were given logic questions that we needed to answer. Here's the first question.
On of Isle of View, a knight always tell the truth and a knave always lie. A person who just came met Mr. A and Mr. B. Mr. A said "one of us is a knave". Which one is the knave, and which one is the night?
We need to answer this question by Proof by Contradiction, which means we have to assume the opposite of the statement and prove that it's right. This was how we tackled this question.
First, we choose the Direct Reasoning of the statement, which in this case was
1) Assume Mr. A is a knight. We can then tell that
2) His statement is true, therefore our conclusion is that
3) Mr. B is a knave and Mr. A is a knight.
Using the Indirect Reasoning, we take the opposite statement and check if it leads to contradiction...
1) Assume Mr. A is a knave
2) His statement is false
3) They are both knights
This part of the reasoning leads to a contradiction, which was what we were looking for. Therefore, Mr. A is a knight.
We looked at two more questions that deals with logic, obviously :) It was pretty much straight forward... We do have homework, however that we need to do. It's Exercise 50 all questions.
Next scribe is Charizze
Pre-test on Monday - January 5, 2009
HAPPY HOLIDAYS!
see you all next year...
see you all next year...
Getting Ready For The Logic Test
A few links you may find useful in preparing for your test on Thursday:
And here are a few quizzes (refresh the page for more) ...
And finally, here are some logic puzzles to practice with ...
Study hard and do your best work; work you're proud of!
- Venn Diagrams
- Conditionals (If ... then ...)
- Biconditionals (... if and only if ...)
- Related Conditionals (the other ones)
And here are a few quizzes (refresh the page for more) ...
- Inductive Reasoning and Conjecture (5 questions)
- Conditional Statements (5 questions)
- Venn Diagrams Self Test (13 questions)
And finally, here are some logic puzzles to practice with ...
Study hard and do your best work; work you're proud of!
Thursday, December 18, 2008
Logic
Today in class we learn about Conditional Statement.
Conditional Statement = is a statement having a "if" and "then".
= it consist of an Hypothesis "if" and a Conclusion "then".
Example: "If two lines are parallel then they do not intersect".
There are 3 Related Conditionals:
1. Converse = is formed by interchanging the hypothesis and conclusion of the original statement.
Ex. "If two lines do not intersect then the lines are parallel.
2. Inverse = is formed by negating the hypothesis and conclusion of the original statement.
= the word "not" is added to both parts of the sentence.
Ex. "If two lines are not parallel then they do intersect.
*note: In algebra we also use Inverse, remember when we subtract positive we get negative and when we add both negative we get positive, well its the same thing in Logic. It is easy to remember Inverse if we know this.
3. Contrapositive = is formed by negating the hypothesis and the conclusion and interchanging the resulting negative.
Ex. "If two lines do intersect then the lines are not parallel.
In class we also talk about biconditional.
Biconditional = it is a logical operator connecting two statements if and only if, where hypothesis and conclusion. Its like they are both going on the same direction.
-more information-
and that`s it for today, don`t forget to do the homework Exercise 49 and finishing the homework from the blog.
next scribe is Jeamille.
Conditional Statement = is a statement having a "if" and "then".
= it consist of an Hypothesis "if" and a Conclusion "then".
Example: "If two lines are parallel then they do not intersect".
There are 3 Related Conditionals:
1. Converse = is formed by interchanging the hypothesis and conclusion of the original statement.
Ex. "If two lines do not intersect then the lines are parallel.
2. Inverse = is formed by negating the hypothesis and conclusion of the original statement.
= the word "not" is added to both parts of the sentence.
Ex. "If two lines are not parallel then they do intersect.
*note: In algebra we also use Inverse, remember when we subtract positive we get negative and when we add both negative we get positive, well its the same thing in Logic. It is easy to remember Inverse if we know this.
3. Contrapositive = is formed by negating the hypothesis and the conclusion and interchanging the resulting negative.
Ex. "If two lines do intersect then the lines are not parallel.
In class we also talk about biconditional.
Biconditional = it is a logical operator connecting two statements if and only if, where hypothesis and conclusion. Its like they are both going on the same direction.
-more information-
Logical Relationships Between Conditional Statement
Watch Video on Conditional Statements
Conditional Statements - Converse and Biconditional
and that`s it for today, don`t forget to do the homework Exercise 49 and finishing the homework from the blog.
next scribe is Jeamille.
Using Venn Diagrams in Word Problems
In class today, we went over some questions on Venn diagram sets. You can see the answers to in the slides that Mr. K posted.
After doing a bit of that, we then learned how to apply our knowledge of Venn diagram sets and rules to problems that you would see in real life.
Let's see how we would solve problems that look like this with Venn diagrams:
"In a group of students 12 are taking chemistry, 10 are taking physics, 3 are taking both and 5 are taking neither. How many students are in the group?"
First we would draw our 'universe' and the Venn diagram that goes in it.:
('U' being universe, 'C' being chemistry, and 'P' being physics)
After you have everything set up, it's time to add in some numbers. Let's start with the easiest possible ones first.
Seeing how there are 5 students who aren't taking either chemistry or physics, we can tell right away that they're going to be placed somewhere in the 'universe', outside of the Venn diagrams.
The 3 students who are taking both courses are the next obvious choice. Since they are taking chemistry AND physics, they belong in the intersection, where the 'C' and the 'P' circle meet.
Now we'll figure out how many students are taking each course.
Automatically your brain might think that the number of students taking chemistry is 12 and the number of students taking physics is 10, but neither statement is true.
The actual number of students taking chemistry is 9, and the number of students taking physics is 7. Why? Because we exclude the 3 students that are taking both courses.
We subtract those 3 students from the pool of students taking chemistry (12 - 3 = 9) and physics (10 - 3 = 7).
We do this because in circles 'C' and 'P', we only want the number of students who are ONLY taking either chemistry or physics, excluding the students that are taking both.
Using the information we've gathered, we can determine the total number of students by adding up all the numbers. You should then finish off with this:
And that's how you solve a word problem with a Venn diagram.
We did a couple more examples, which all use pretty much the same procedure.
We did however, do one that involved 3 circles in the Venn diagram instead of 2. It seemed a little confusing, but solving it involved the same procedures as the previous examples. You just had to be very careful with your numbers.
That's about it for my scribe post, thanks for reading. The next scribe will be..camiLLe
After doing a bit of that, we then learned how to apply our knowledge of Venn diagram sets and rules to problems that you would see in real life.
Let's see how we would solve problems that look like this with Venn diagrams:
"In a group of students 12 are taking chemistry, 10 are taking physics, 3 are taking both and 5 are taking neither. How many students are in the group?"
First we would draw our 'universe' and the Venn diagram that goes in it.:
('U' being universe, 'C' being chemistry, and 'P' being physics)
After you have everything set up, it's time to add in some numbers. Let's start with the easiest possible ones first.Seeing how there are 5 students who aren't taking either chemistry or physics, we can tell right away that they're going to be placed somewhere in the 'universe', outside of the Venn diagrams.
The 3 students who are taking both courses are the next obvious choice. Since they are taking chemistry AND physics, they belong in the intersection, where the 'C' and the 'P' circle meet.
Now we'll figure out how many students are taking each course.Automatically your brain might think that the number of students taking chemistry is 12 and the number of students taking physics is 10, but neither statement is true.
The actual number of students taking chemistry is 9, and the number of students taking physics is 7. Why? Because we exclude the 3 students that are taking both courses.
We subtract those 3 students from the pool of students taking chemistry (12 - 3 = 9) and physics (10 - 3 = 7).We do this because in circles 'C' and 'P', we only want the number of students who are ONLY taking either chemistry or physics, excluding the students that are taking both.
Using the information we've gathered, we can determine the total number of students by adding up all the numbers. You should then finish off with this:
And that's how you solve a word problem with a Venn diagram.We did a couple more examples, which all use pretty much the same procedure.
We did however, do one that involved 3 circles in the Venn diagram instead of 2. It seemed a little confusing, but solving it involved the same procedures as the previous examples. You just had to be very careful with your numbers.
That's about it for my scribe post, thanks for reading. The next scribe will be..camiLLe
Wednesday, December 17, 2008
Sets and Venn Diagrams
Below are some definitions we learn't in class today.
SET: a collection of objects, may be things, people, numbers, etc.
SUBSET: a "smaller" collection of objects taken from a given set.
SET: a collection of objects, may be things, people, numbers, etc.
SUBSET: a "smaller" collection of objects taken from a given set.
UNIVERSE: the set of all objects being considered.
NULL SET: a set that is empty, a set with no members, objects, etc.
NB: If two sets have no common elements, they are said to be "disjoint".
OPERATION ON SETS
OPERATION ON SETS
Below are some terms and examples
Union ( U )
A U B means gather all the objects in a set A with all the objects in set B.
Union ( U ) means "or".
A U B means gather all the objects in a set A with all the objects in set B.
Union ( U ) means "or".
Exclusion ( / )
A/B means everything that is in set A excluding those things that are also in set B.
Exclusion ( / ) means "excluding".
A/B means everything that is in set A excluding those things that are also in set B.
Exclusion ( / ) means "excluding".
Intersection ( n )
A n B means to gather all the objects that are in both set A and B.
Intersection ( n ) means "and".
A n B means to gather all the objects that are in both set A and B.
Intersection ( n ) means "and".
Do your homework ... Practise makes Perfect!
Good luck to all!
The next scribe is Eric.
Tuesday, December 16, 2008
Monday, December 15, 2008
logic
After that we finished the geometry unit. Today we started a new unit called logic.We started by looking at what the word "logic" means. we then watched a videos about the word argument, what it means on http://www.youtube.com/watch?v=teMlv3ripSM
We later studied some important words and definitions in logic
Logic: (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration. en.wikipedia.org/wiki/Logic
The argument: A fact or statement used to support a proposition; a reason:; A verbal dispute; a quarrel; A process of reasoning; A series of statements organized so that the final statement is a conclusion which is intended to follow logically from the preceding statements, which function as premises... en.wiktionary.org/wiki/argument .
The argument consists of two (2) parts:
1. a set of premises
2. a conclusion
We then started to make a difference between premises and a conclusion, valid argument and invalid argument
A premise is a valid truth
A conclusion is a result of an argument
An argument is called valid if its result is direct consequence of its premises. In a valid argument, if the premises are true, then the conclusion must also be true.
An argument is called invalid when its conclusion is not a direct consequence of it’s premises even if its conclusion are true.
Sound argument: an argument is sound if it is both valid and true
After learning the validity and the invalidity of an argument we learned the induction and deduction.
Induction: when we observe several particular examples of same things, and then conjecture that the “the discovered” pattern must always be true
Deduction: a process of reasoning in which a conclusion follows necessarily from the premises presented, so that the conclusion cannot be false if the premises are true.
We later did a drag and drop exercise on valid argument and invalid argument, inductive and deductive reasoning.
Mr. K gave us a homework which exercise number 45
The next scribe is Zawadi.
We later studied some important words and definitions in logic
Logic: (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration. en.wikipedia.org/wiki/Logic
The argument: A fact or statement used to support a proposition; a reason:; A verbal dispute; a quarrel; A process of reasoning; A series of statements organized so that the final statement is a conclusion which is intended to follow logically from the preceding statements, which function as premises... en.wiktionary.org/wiki/argument .
The argument consists of two (2) parts:
1. a set of premises
2. a conclusion
We then started to make a difference between premises and a conclusion, valid argument and invalid argument
A premise is a valid truth
A conclusion is a result of an argument
An argument is called valid if its result is direct consequence of its premises. In a valid argument, if the premises are true, then the conclusion must also be true.
An argument is called invalid when its conclusion is not a direct consequence of it’s premises even if its conclusion are true.
Sound argument: an argument is sound if it is both valid and true
After learning the validity and the invalidity of an argument we learned the induction and deduction.
Induction: when we observe several particular examples of same things, and then conjecture that the “the discovered” pattern must always be true
Deduction: a process of reasoning in which a conclusion follows necessarily from the premises presented, so that the conclusion cannot be false if the premises are true.
We later did a drag and drop exercise on valid argument and invalid argument, inductive and deductive reasoning.
Mr. K gave us a homework which exercise number 45
The next scribe is Zawadi.
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