What’s something new I learned this week?
1、Translate “unless" as “if not":
When we see "unless", we should translate it as "if not". For example,
"Don't knock it unless you've tried it" which could be translated as
"don’t knock it if you don’t tried it", which equals "tried it, knock it
(as learned last week: P ⇒ Q ⇔ ¬Q ⇒ ¬P )
When we see "unless", we should translate it as "if not". For example,
"Don't knock it unless you've tried it" which could be translated as
"don’t knock it if you don’t tried it", which equals "tried it, knock it
(as learned last week: P ⇒ Q ⇔ ¬Q ⇒ ¬P )
2、 Conjunction: ∧
Combine two statements by claiming they are both true with logical
“and":
A(x ) and B(x ) (python keyword and works like this)
A(x )∧ B(x ) (^ is a symbol for "and")
As sets: x ∈ A ∩B
Notice that a conjunction is false if either part is false.
3、 disjunction: ∨
Combine two statements by claiming that at least one of them
is true using or ( ∨in symbols).
A(x ) or B(x ) (the python keyword or works like this)
A(x )∨ B(x ) (in symbols)
x ∈A ∪ B (as sets)
4、 Notice the close connection between the symbols for
conjunction and intersection,∧ , ∩, and the symbols for
disjunction and union, ∨, ∪. Coincidence? In any case, you
may use it as a mnemonic.
5、 Negation:¬
5、 Negation:¬
Negate the statement "All employees earning over 110,000 are
female." Usually prepending the word "Not" will work, and in
logic we use the corresponding symbol
A good exercise is to "work" the negation : as far into the
statement as possible. The statement is true exactly when its
negation is false.
Put "Not" in the front of the sentence.
Put "Not" in the front of the sentence.
6 、Negating implications is a common task. There are several
equivalent ways of doing this, but some are more common than
others. Try negating the following in such a way that the :
symbol applies to the \smallest possible" part of the expression:
8x 2 X;P(x ) ) Q(x )
Now for symmetry, negate the following in such a way that the
: symbol applies to the \smallest possible" part of the
expression:6
9x 2 X;P(x ) ^ :Q(x )
7、 Standard negation
Negated expressions have some standard transformations:
I :(P(x ) ) Q(x )) , P(x ) ^ :Q(x )
I :(P(x ) ^ Q(x )) , P(x ) ) :Q(x ) (has this become
asymmetrical?)
8、 There's no reason that the y in the antecedent would be
the same as the y in the consequent. It could be re-written:
(8x 2 R; 9y 2 R; x < y) ) (8z 2 R; 9w 2 R; z 2
< w)
9、 A predicate is a function that takes in a value and returns a
9、 A predicate is a function that takes in a value and returns a
boolean (T or F).
What’s something that frustrated and challenged me this week?
1 Bi-implication
2 The difference between implication and conjunction
At first, I simply cannot figure out what's the difference between two of them, so I approached to my TA, Christina for help, and she did help me a lot, here is something I think is quite significant in her email:
......There are 3 ways to show that they are different:
1) Their truth tables:
P Q P ^ Q P -> Q
T T T T
T F F F
F T F T
F F F T
Did you study truth tables in class? If you need more elaboration, feel free to ask me. :)
Clearly, from their truth tables, they are different.
2) Compare P ^ Q to P -> Q 's alternate form: ¬P v Q
So another way of writing P -> Q is ¬P v Q, right? ¬P v Q and P ^ Q are definitely not the same.
3) Compare their English meaning:
P -> Q means "If P, then Q". In other words, this expression is sayying "if P is true, then Q is true".
P ^ Q means "P and Q". In other words, this expression is saying "P and Q are both true".
Sometimes... It might be easier if you replace your symbols with actual meaning. So think of P as "Cats are fat" and Q as "Dogs are fat".
P -> Q is then saying, "If cats are fat, then dogs are fat"
and P ^ Q is saying "cats are fat and dogs are fat"
...These two statements have different meanings right?
2 The difference between implication and conjunction
At first, I simply cannot figure out what's the difference between two of them, so I approached to my TA, Christina for help, and she did help me a lot, here is something I think is quite significant in her email:
......There are 3 ways to show that they are different:
1) Their truth tables:
P Q P ^ Q P -> Q
T T T T
T F F F
F T F T
F F F T
Did you study truth tables in class? If you need more elaboration, feel free to ask me. :)
Clearly, from their truth tables, they are different.
2) Compare P ^ Q to P -> Q 's alternate form: ¬P v Q
So another way of writing P -> Q is ¬P v Q, right? ¬P v Q and P ^ Q are definitely not the same.
3) Compare their English meaning:
P -> Q means "If P, then Q". In other words, this expression is sayying "if P is true, then Q is true".
P ^ Q means "P and Q". In other words, this expression is saying "P and Q are both true".
Sometimes... It might be easier if you replace your symbols with actual meaning. So think of P as "Cats are fat" and Q as "Dogs are fat".
P -> Q is then saying, "If cats are fat, then dogs are fat"
and P ^ Q is saying "cats are fat and dogs are fat"
...These two statements have different meanings right?
How does course material relate to other classes or interests?
1 De Morgan's Law(s)
That's what I've learned before in mat137 class.
2 Some laws:
Commutative laws, Associative laws and Distributive laws I've seen before in my high school.
Commutative laws, Associative laws and Distributive laws I've seen before in my high school.
How did your tutorial/test/assignment go this week?
Tutorial Problems for week3.
Problem solving:
Friday Class Folding
Understanding problem:
Right hand hold on the paper and left hand fold the paper. If fold the paper n times, predicting the sequence of fold pattern.
Design a plan:
fold a strip of paper four times,observe the sequence of fold pattern and make record from each folded form. Then predict the 5th fold and testify it.
Carrying out a plan:
make a folded form: n - the number of the total fold, U - up fold point, D - down fold point first to know the relation between number of folds and folded area. A - number of folded area, m - the number of fold point(up or down)
we can get the sequence of ups and downs for any nth fold from above form: if I need fold pattern of n=3, I can start from n=2,first copy the letters "d" or "u" of the 2nd row to the 3rd row, the on the 3rd row, put "d" on the rightest. Now using opposite characteristic I get a "u" which is on the other side of the "d". then according to the above analysis, I can get new fold line of the n=3 row.
Understanding problem:
Right hand hold on the paper and left hand fold the paper. If fold the paper n times, predicting the sequence of fold pattern.
Design a plan:
fold a strip of paper four times,observe the sequence of fold pattern and make record from each folded form. Then predict the 5th fold and testify it.
Carrying out a plan:
make a folded form: n - the number of the total fold, U - up fold point, D - down fold point first to know the relation between number of folds and folded area. A - number of folded area, m - the number of fold point(up or down)
we can get the sequence of ups and downs for any nth fold from above form: if I need fold pattern of n=3, I can start from n=2,first copy the letters "d" or "u" of the 2nd row to the 3rd row, the on the 3rd row, put "d" on the rightest. Now using opposite characteristic I get a "u" which is on the other side of the "d". then according to the above analysis, I can get new fold line of the n=3 row.
