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THE SEED MONTESSORI SCHOOL GENERAL MATHEMATICS 11 Operations on Propositions July 14, 2016
THE SEED MONTESSORI SCHOOL
GENERAL MATHEMATICS 11
Operations on Propositions
July 14, 2016
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Daily Routine
Objectives
Recall
Lesson Proper
Practice Exercises
Exit Card
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On ¼ sheet of yellow paper, answer item numbers 1-5 of Gear Up on page 198.
Time Limit: 5 minutes
At the end of the period, you are expected to be able to:
perform the different operations on propositions O
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Give one example of a compound proposition that does not make sense.
Example:
Let:
r = it rains
w = the soil gets wet
d = President Duterte gets impeached
S1 : if r, then w
S2 : if r, then d
S1 makes sense while S2 does not.
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Logical operators – words used to connect 2 or more propositions to form compound propositions
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LOGICAL OPERATORS
1. Negation ( ~ )
2. Disjunction ( v )
3. Conjunction ( ^ )
4. Implication ( )
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The negation of a proposition is its exact opposite. It is denoted by the symbol ~.
Example:
Let:
P: 1 is an even number
Its negation is represented by ~P (read as “not P”) and is written as
~P: 1 is not an even number
In simple English, we say that
~P: 1 is an odd number
This is because of the dichotomy of evenness and oddness of numbers.
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An event is said to be dichotomous if exactly one of two possible instances can happen at a time.
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DICHOTOMOUS EVENTS
NON-DICHOTOMOUS EVENTS
head or tail (in a single coin toss)
prime or composite (numbers); 1 is neither prime nor composite
on and off (lights) citizenship
even or odd (numbers) walking and thinking
equal or unequal
Let P be a proposition. Other simple English translations of the negation of P are as follows:
not P
it is not the case that P
it is false that P
it is not true that P
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Negate each proposition. Express your answer in simple English.
1. A: I am under the STEM strand.
~A: I am not under the STEM strand.
2. B: The square of a negative number is positive.
~B: It is not the case that the square of a negative number is positive.
3. C: I need you more than I need oxygen.
LOGICAL OPERATORS
1. Negation ( ~ )
2. Disjunction ( v )
3. Conjunction ( ^ )
4. Implication ( )
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The disjunction of two (or more) propositions is a compound proposition whose truth or falsity depends on the propositions themselves.
Example:
Let:
P: 2 is a prime number
Q: 2 is an even number
Their disjunction is represented by P v Q (read as “P or Q”) and is written as
P v Q: 2 is a prime or an even number
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Inclusive or: one or the other, or both
I will eat hamburger or fries.
Exclusive or: one or the other, but not both.
A single toss of a coin will show a head or a tail.
Unless otherwise stated, all disjunctions are inclusive by default.
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Let P and Q be propositions. Some simple English translations of the disjunction of P and Q are as follows:
P or Q
either P or Q
P unless Q
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Write the disjunction of the given propositions. Express your answer in simple English.
1. D: Yohan will take the Academic Track.
E: Yohan will take the Tech-Voc Track.
D v E: Yohann will take the Academic Track unless he takes the Tech-Voc Track.
2. F: Give me another chance.
G: I will never love again.
F v G: Give me another chance or I will never love again.
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LOGICAL OPERATORS
1. Negation ( ~ )
2. Disjunction ( v )
3. Conjunction ( ^ )
4. Implication ( )
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The conjunction of two propositions implies the idea of “both” propositions taken at the same time.
Example:
Let:
P: 2 is a prime number
Q: 2 is an even number
Their conjunction is represented by P ^ Q (read as “P and Q”) and is written as
P ^ Q: 2 is a prime and an even number
In simple English, we say that
2 is an even prime number.
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Let P and Q be propositions. Some simple English translations of the conjunction of P and Q are as follows:
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• P and Q • P moreover Q • P although Q • P still Q • P furthermore Q
• P also Q • P nevertheless Q • P however Q • P yet Q • P but Q
Write the conjunction of the given propositions. Express your answer in simple English.
1. H: The solutions of x2 + 6x + 9 = 0 are equal.
J: The solutions of x2 + 6x + 9 = 0 are rational.
H ^ J: The solutions of x2 + 6x + 9 = 0 are equal and rational.
2. K: My brain says stop.
L: My heart says go on.
K ^ L: My brain says stop but my heart says go on.
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LOGICAL OPERATORS
1. Negation ( ~ )
2. Disjunction ( v )
3. Conjunction ( ^ )
4. Implication ( )
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An implication is also called a conditional statement or an if-then statement. The “if” part is called the antecedent (or hypothesis or premise) while the “then” part is called the consequent (or conclusion).
Example:
Let:
S: I get a perfect score in the exam.
T: I will treat you to lunch.
The implication is represented by ST (read as “if S, then T”) and is written as
ST : If I get a perfect score in the exam, then I will treat you to lunch.
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Implications are directional. If the “forward” is true, the “backward” is not always true.
Case 1: Backward is TRUE
Example:
P: The sum of two angles is 90°.
Q: Two angles are complementary.
PQ: If the sum of two angles is 90°, then they are complementary.
The statement, “If two angles are complementary, then their sum is 90°.” is also true.
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Implications are directional. If the “forward” is true, the “backward” is not always true.
Case 2: Backward is FALSE
Example:
P: It rains.
Q: The soil gets wet.
PQ: If it rains, then the soil gets wet.
However, the soil being wet does not necessarily mean that it rained. There might have been other factors that made the soil wet.
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Let P and Q be propositions. Some simple English translations of the implication of P to Q are as follows:
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• if P then Q • P implies Q • P is a sufficient condition for Q • Q is a necessary condition for P
• Q if P • Q follows from P • Q provided P • Q whenever P • Q is a logical consequence of P
Write an implication given the propositions. Express your answer in simple English.
1. M: All the angles of a triangle are acute.
N: A triangle is acute.
MN: If all the triangles of a triangle are acute, then the triangle is acute.
2. O: You think that I’m still holding on.
P: You should go and love yourself.
OP: If you think that I’m still holding on, then you should go and love yourself.
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Parentheses, brackets and braces are called grouping marks. The following are the guidelines in using them when logical operators are involved.
1. “both” – “and” and “either” – “or”
* Insert grouping marks right after either or both.
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Propositional Phrase Propositional Form
both P or Q and R (P v Q) ^ R
P or both Q and R P v (Q^R)
either P and Q or R (P^Q) v R
P and either Q or R P ^ (Q vR)
Parentheses, brackets and braces are called grouping marks. The following are the guidelines in using them when logical operators are involved.
2. “neither-nor”
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Propositional Phrase Propositional Form
neither P nor Q
~ ( P v Q ) not either P or Q both P or Q are not
Parentheses, brackets and braces are called grouping marks. The following are the guidelines in using them when logical operators are involved.
3. “both not” and “not both”
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Propositional Phrase Propositional Form
P and Q are not both ~ ( P ^ Q )
P and Q are both not ~P ^ ~Q
Answer the following on ¼ sheet of yellow paper.
Let: P: I am walking.
Q: I am running.
R: I am jogging.
1. ~P
2. ~P ^ Q
3. ~(P v Q)
4. ~R P
5. ~(Q v R)
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Let: P: I am walking.
Q: I am running.
R: I am jogging.
1. ~P: I am not walking.
2. ~P ^ Q: I am not walking but running.
3. ~(Q v P): I am neither running nor walking.
4. ~R P: If I am not jogging, then I am walking.
5. ~(P ^Q): I am not both walking and running.
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Give one compound proposition that is related to your strand.
STEM – If a star explodes, then a black hole will be formed and it will consume everything that goes nearby.
ABM – If it is the 15th or the 30th of the month, then it is good to put some products on sale.
HUMSS – Both the Philippines and China may not prohibit each other in fishing in the West Philippine Sea if both countries are to abide by the decision of the International Tribunal.
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