Math 10 Chapter 1 Lesson 1: Sentences
1. Summary of theory
1.1. Clause. The clause contains the variable.
a) Proposition

A statement cannot be both true and false.

A true statement is called a true statement. A false statement is called a false proposition.
Eg:
“4 is a composite number” is a true statement.
27 is not divisible by 9 is false.
b) A clause containing a variable
Eg: Consider the sentences:
(a): “x + 1 = 2”
(b): “n is a natural number”
Find two values of x, n so that (a), (b) get one true statement and one false statement.
Sentences (a) and (b) are examples of clauses containing variables.
1.2. Negation of a proposition
– The notation of the negative clause of the proposition P is \(\overline P \), we have :
\(\overline P \) is true when P is false.
\(\overline P \) is false when P is true.
Eg:
– For the proposition P: “\(\pi \) is an integer”. We have: \(\overline P :\) “\(\pi \) is not an integer”.
– For proposition Q: “In a triangle, the side opposite the larger angle is larger”.
– We have: \(\overline Q :\) “In a triangle, the side opposite the larger angle is smaller”.
1.3. The following clause
Eg: Consider the form of the statement “If the northeast monsoon returns, it will become cold”.
– The proposition “If P then Q” is called the following proposition and is denoted by \(P \Rightarrow Q\).
– The clause \(P \Rightarrow Q\) is false only when P is true Q is false.
– Mathematical clauses are usually of the form \(P \Rightarrow Q\)

P is the hypothesis, Q is the conclusion of the theorem.

Either P is a sufficient condition for Q, or Q is a necessary condition for P.
Eg: Given the proposition: “If triangle ABC has three equal sides, then triangle ABC is an equilateral triangle”.
1.4. Inverse clause – Two equivalent clauses
Eg: Given the real number x. Consider:
P: “x is a rational number”.
Q: “2x is a rational number”.
a) State the statements \(P \Rightarrow Q\) and \(Q \Rightarrow P\).
b) Consider the true and false of the two statements \(P \Rightarrow Q\) and \(Q \Rightarrow P\).
We have:
 \(P \Rightarrow Q\): “If x is a rational number, then 2x is a rational number”. (Correct)
 \(Q \Rightarrow P\): “If 2x is a rational number, then x is a rational number”. (Correct)
Define:

The clause \(Q \Rightarrow P\) is called the inverse of the clause \(P \Rightarrow Q\).

If both the statements \(P \Rightarrow Q\) and \(Q \Rightarrow P\) are true, then we say that P and Q are equivalent and denoted \(P \Leftrightarrow Q\).
Reading convention:
1.5. Sign \(\forall \) and \(\exists\).
Eg: Given the following clauses:
P: “Every even number is divisible by 2”.
Q: “There is a rational number less than its reciprocal”.
State the negation of the above statements. Consider the true and false of the statements P, Q, \(\overline P \), \(\overline Q \).
We have:
 \(\overline P :\) “There is an even number that is not divisible by 2”.
 \(\overline Q :\) “Every rational number is greater than or equal to its reciprocal”.
 P is true, \(\overline P \) is false
 Q is true, \(\overline Q \) is false, for example \(\frac{1}{3} < 3\).
– The symbol \(\forall \) reads as “for all”.
– The symbol \(\exists \) reads as “there is one” (there exists one) or “there is at least one”.
Comment:

The negation of \(”\forall x \in X,P(x)”\) is \(”\exists x \in X,\overline {P(x)} ”.\)

The negation of \(”\exists x \in X,P(x)”\) is \(”\forall x \in X,\overline {P(x)} ”.\)
Eg:
Pclause: “\(\exists n \in \mathbb{N}:{n^2} = n\)”
There exists a natural number n whose square is equal to itself.
For all integers:
Qclause: “\(\forall x \in \mathbb{Z}:{x^2} = x\)”
The square of every integer x is equal to itself.
2. Illustrated exercise
Question 1: Consider whether the following statements are propositions? If it is a proposition, what is true or false?
a) \(\sqrt 3 \) is not an integer.
b) Quoc is an Asian country, right?
c) The equation \({x^2} 4x + 3 = 0\) has no solution.
d) That dog is so cute!
e) 2x + 3 is a positive number.
f) If n is odd, then n is divisible by 3.
g) If n is divisible by 2 then n is even.
Solution guide
a) This is a true statement.
b) This is a question, not a proposition.
c) This is a false statement because the equation has a solution x=1.
d) This is an exclamation, not a clause.
e) This is a false statement because if x= 2 exists then 2x + 3 = 1 is negative.
f) This is a false statement because n=1 is odd but not divisible by 3.
g) This is a true statement.
Verse 2: Find the inverse of the following statements and state whether this inverse is true or false: “If the line is parallel, then the two lines have nothing in common”.
Solution guide
The given proposition has the form: \(P \Rightarrow Q\) where P is “two parallel lines”, Q is “two lines that have no common ground”. So the inverse proposition is: “If two lines have no point in common, then the two lines are parallel”. This statement is correct.
3. Practice
3.1. Essay exercises
Question 1: Consider whether the following statements are propositions? If it is a proposition, what is true or false?
a) \({x^2} + 2x + 2\) is a negative number.
b) If n is even, then n is divisible by 2.
c) If n is divisible by 8, then n is even.
d) n is odd if and only if \(n^2\) divisible by 3.
Verse 2: Find the inverse of the following statements and state whether this inverse is true or false: “An isosceles triangle with an angle equal to 60 degrees is an equilateral triangle”.
Question 3: Find the negative statements of the following statements and say whether they are true or false.
a) \(P = ”\forall x \in R,{x^2} \ge 0”.\)
b) Q = “There is a quadrilateral with no angle less than ”.
3.2. Multiple choice exercises
Question 1: Which of the following statements is incorrect?
A. Two triangles with equal areas are congruent.
B. Two similar triangles have the same area.
C. A triangle with three equal sides has three equal angles.
D. A triangle with three equal angles has three equal sides.
Verse 2: Which of the following statements is correct?
A. \(\exists n \in N,n < 0\)
B. \(\exists n \in Q,{x^2} = 2\)
C. \(\forall x \in R,{x^2} – x + 1 > 0\)
D. \(\forall x \in Z,\frac{1}{x} > 0\)
Question 3: The clause \(\forall x \in R,{x^2} – 2 + a > 0\) where a is a given real number. Find a so that the clause is correct
A. a < 2
B. \(a\le2\)
C. a = 2
D. a > 2
Question 4: Which of the following statements is correct?
A. \(\forall x \in \mathbb{N}:\) x is divisible by 3
B. \(\exists x \in \mathbb{R}:{x^2} < 0\)
C. \(\forall x \in \mathbb{R}:{x^2} > 0\)
D. \(\exists x \in \mathbb{R}:x > {x^2}\)
Question 5: Which of the following statements is false?
A. A triangle is a right triangle if and only if it has one angle equal to the sum of the other two angles.
B. A triangle is equilateral if and only if it has two equal medians and an angle equal to \({60^0}.\)
C. Two triangles are congruent if and only if they are similar and have a congruent side.
D. A quadrilateral is a rectangle if and only if it has 3 right angles.
4. Conclusion
Through this lesson, you should achieve the following goals:
 Understand the most basic content of the most basic concept of the proposition.
 Identify the inverse and negative clauses of a proposition.
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