MATHEMATICS CHAPTER 10

1. Chapter 10
Logic

2. lesson 36
PropositionS
A proposition is a declarative
sentence that is either TRUE or
FALSE,but not both . If a proposition is
true,then its truth value is true which
is denoted by T; otherwise,its true
value is false and is denoted by F.

3. EXAMPLE 1. Determine whether each of the following
statement is a proposition or not. If a proposition,give its true
value.
p:Mindanao is an isalnd in the Philippines.
Q: Find a number which devides your age.
R:My seatmate will get a perfect score in the logic exam.
S:Welcome to the philippines!
T:3+2=5

4. ,you are a FilipinoU: (x)=
𝑥
𝑥+1
is a rational function.
V: what is a domain of the function?
W: I am lying.
P1:it is not case that 2 is a rational number.
P2: Either logic is a fun and interesting,or it is boring.
P3: if you are a grade 11 studenthen .

5. P4: if you are more than 60 years old,then you are
entitled to a senior citizen’s card,and if you are
entitled to a senior citizen’s card,then you are more
than 60 years old.

6. Solution:Recall that for a statment to be
proposition it has to be a declarative
sentens,and it should have a truth value of
either true or false ,but not both true and
false at the same time,
p.This is declarative sentence,and Mindanao
is an island in the Philippines.Hence,p is a
true proposition
q.This is an imperative sentence,and so it is
not a proposition.

7. r. The statement is a declarative
sentence.Although the truth value will only
be known after the logic exam,we know that
it can only be either true(my seatmate gets a
perfect score)or false (she has some
mistakes),but not both.Hence,are is a
proposition.
s.Statement s is an exclamatory sentence,
and so it is not a proposition.
t.Obviously,3 + 2 = 5 is a true mathematical
sentence.

8. DEFINITION
Compound proposition is a proposition
formed from simpler propositions using
logical connectors or some combination of
logical connectors.
A proposition is simple if it cannot be broken
down any futher into other component
propositions.

9. EXAMPLE 2
For each of the proposition in example 1,
determine whether it is a simple or
compound proposition
Solution.the propositions p, r, t and u are all
simple proposition.
P1: It is not the case that 2 is a rational
number.
P2:Either logic is a fun and interesting, or it
is boring.

10. LESSON 37:LOGICAL OPERATORS
DEFINITION
The negation of a proposition p is denoted by
~p:(read as `not`p)
P~p
T F
F T
Solution. The negation of the given proposition are given below.
~n1:it is not true that 2is an odd,or`2 is an even number.`

11. Definition.
The negation of a proposition P is denoted by
~p: (read as “not” p,)
and is defined through its tuth table:
P ~p
T F
F T

12. EXAMPLE 2. State the negation of the following
propositions.
n1: p(x) =
𝑥−1
𝑥+2
is polynomial function.
n2: 2 is an odd number.
N3: The tinikling is the most difficult dance.
N4: everyone in visayas speaks cebuano.

13. solution. The negation of the given propositons are given
below.
~n1: it is not true that p (x) =
𝑥−1
𝑥+2
is polynomial
function , or more simply, p(x) =
𝑥−1
𝑥+2
is not a polynomial
function,.
~n2: it is not true that 2 is an odd number’, or 2 is an
even number
~n3: The tinikling is not a most difficult dance’

14. EXAMPLE 8. Determine the truth values of the
following propositions.
(a) if 2>0,then there are more than 100 million
Filipinos.
(b) if2>0, then there are only 5 languages spoken
in the philippines.
(c) if 2>0, then it is more fun in the Philippines.

15. SOLUTION. The number 2 is a positive number,and
so the proposition ‘2>0’ is true, while ‘2<0’ is false.
(a) The hypothesis and the conclusion are both
true. Hence the conditional is true.
(b) The hypothesis is true but the conclusion is
wrong because there are more than 5 languages in
the Philippines! In fact there are more than 100
languages in the country. Thus ,the conditional is
false.

16. (c) Because the hypothesis is false, the
conditional is true whether it is indeed
more fun in the Philippines or not .

17. LESSON 38: CONSTRUCTING TRUTH
TABLES
Learning outcome: at the end of the
lesson,the learner is able to determined
the possible truth values of propositions.

18. EXAMPLE 1. since a propositions has two
possible truth values, propositions P would
have the following truth table.
The truth table is useful because we can use it to display all the possible truth value
combinations of two or more propositions.for example, suppose P and Q are propositions . We
can construct a truth table displaying the relationship between the possible truth values of P
and the truth values of Q. The rows of the table will correspond to each of the possible truth
value combination of P and the Q, and so there will be 22
= 4 rows.
P
T
F
p q
T T
T F
F T
F F

19. Definition.
The negation of a proposition P is denoted by
~p:(read as ‘not’ p,)
and is defined through its truth table
P ~P
T F
F T

20. Definition.
A proposition that is always true is called a
TAUTOLOGY, while a proposition that is always false
is called a CONTRADICTION. A tautology is denoted
by r and contradiction by ∅.
EXAMPLE 1. Let p and q be propositions. Using truth
tables , show the following
i. p V r is tautology
ii. P ∧ ∅ is a contradiction.
iii.P → ( p V q) is a tautology

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