Download NOTES: Introduction to Logic KEY Definitions

NOTES: Introduction to Logic KEY
Definitions:
Conjecture:
An unproven statement based on observations
Counterexample:
An example that shows a conjecture is false
Inductive Reasoning:
The process of finding a pattern for specific cases then writing a
conjecture for the general case.
Conditional Statement: A logical statement with two parts, a hypothesis and conclusion.
Hypothesis:
The “if” part of a conditional statement
Conclusion:
The “then” part of a conditional statement.
Complete the conjecture based on the pattern you observe.
1.
1+1=2
3+5=8
7 + 11 = 18
21 + 39 = 60
183 + 415 = 598
Conjecture: The sum of any two odd numbers is even.
2.
1x1=1
3 x 5 = 15
7 x 11 = 77
21 x 39 = 819
183 x 415 = 76,945
Conjecture: The product of any two odd numbers is odd.
Sketch the next figure in the pattern.
3.
4.
Predict the next number in the sequence of numbers.
5.
1, 4, 7, 10, 13
6. 123, 234, 345, 456, 567
7. 1.1, 1.01, 1.001, 1.0001, 1.00001
Show the conjecture is false by finding a counterexample.
8.
The sum of two numbers is always greater than the larger number. -5 + 2 = -3
9.
All prime numbers are odd. 2
10. The quotient of two whole numbers is a whole number. 12 ÷ 15
11. The square root of a number x is always less than x. 1
12. If the product of two numbers is positive, then the two numbers must be positive.
(-2)(-1)=2
Underline the hypothesis and circle the conclusion in each of the following examples.
EXAMPLE 1: If it is Tuesday, then Phil plays tennis.
EXAMPLE 2: If you work for 8 hours, then you work for 1/3 of the day.
EXAMPLE 3: Put the following in if – then form.
An angle of 40° is an acute angle.
IF – THEN form If an angle measures 40°, then the angle is acute.
In the following underline the hypothesis and circle the conclusion. If necessary, write
the sentence as a conditional in if-then form.
a) All Olympic competitors are athletes. If you compete in the Olympics, then you are an
athlete.
b) If a polygon is regular, then it is equiangular.
c) All equilateral triangles are isosceles. (Begin “if a figure is . . .”)
If a figure is an equilateral triangle, then it is isosceles.
d) If it doesn’t snow, then we will have school.
Sometimes conditionals can be written using the letter p and q. ‘p’ represents the hypothesis
and ‘q’ represents the conclusion
EXAMPLE 4: if p then q
Symbol p → q
CONVERSE of a conditional
To write the converse of a conditional
1. Identify the hypothesis and conclusion
2. Interchange the hypothesis and conclusion.
If the original conditional is “If p then q.” ( p → q), then the converse is “If q then p”. (q → p)
Write the converse of the following true conditionals.
EXAMPLE 5:
CONDITIONAL: If a polygon is a quadrilateral, then it has four sides.
CONVERSE: If a polygon has 4 sides then it is a quadrilateral.
Write the converse of the true conditional then determine if the converse is also true. If
it is false, then give a counterexample.
EXAMPLE 6:
CONDITIONAL: If a number is divisible by 10, then it is divisible by 5.
CONVERSE: If a number is divisible by 5, then it is divisible by 10.
True / False false
COUNTEREXAMPLE 15
BICONDITIONALS
When a conditional and its converse are true you can combine them as a biconditional.
CONDITIONAL: If a polygon is a quadrilateral, then it has four sides. TRUE
CONVERSE: If a polygon has four sides, then it is a quadrilateral. TRUE
Both the conditional and its converse are true and we can be combined into a biconditional.
BICONDITIONAL: If a polygon is a quadrilateral, then it has four sides and if a polygon has
four sides, then it is a quadrilateral.
We can shorten the biconditional by using if – and – only – if.
IF-AND-ONLY-IF: A polygon is a quadrilateral if and only if it has 4 sides.
NOTES: INVERSE and CONTRAPOSITIVE
The inverse of a statement has the opposite meaning.
EXAMPLE 1:
An angle is acute
An angle is NOT acute
Statement
Negation
Write the negation of each statement.
a) Two angles are vertical.
Two angles are not vertical
b) Two lines are not parallel.
Two lines are parallel
Definition
INVERSE: The inverse of a conditional negates both the hypothesis and conclusion.
EXAMPLE 2:
CONDITIONAL: If a figure is a square, then it is a rectangle.
INVERSE: If a figure is not a square, then it is not a rectangle.
Write the inverse of each statement.
c) If a quadrilateral is a parallelogram, then it has two pairs of parallel sides.
INVERSE: If a quadrilateral is not a parallelogram, then it does not have two pairs of
parallel sides.
d) If you live in a country that borders the United States, then you live in Canada.
INVERSE: If you do not live in a country that borders the United States, then you do not
live in Canada.
Definition
COUNTRAPOSITIVE The contrapositive Interchanges and negates the hypotheses and
conclusion
Write the inverse and contrapositive of the following conditional.
EXAMPLE 3: If you eat all your vegetables, then you will grow.
INVERSE: If you do eat your vegetables, then you will not grow.
CONTRAPOSITIVE: If you did not grow, then you didn’t eat all your vegetables.
Write the inverse and contrapositive of each of the following conditionals.
e) If a triangle is a right triangle, then it has a 90° angle.
INVERSE: If a triangle is not a right triangle, then it does not have a 90° angle.
CONTRAPOSITIVE: If a triangle does not have a 90° angle, then it is not a right triangle.
f) If two segments are congruent, then they have the same length.
INVERSE: If two segments are not congruent, then they do not have the same length.
CONTRAPOSITIVE: If two segments don’t have the same length then they aren’t
congruent.
SUMMARY OF IF / THEN
STATEMENTS
CONDITIONAL
If p then q
Symbol p → q
CONVERSE
If q then p
Symbol q → p
INVERSE
If ∼p then ∼q
Symbol ∼p → ∼q
CONTRAPOSITIVE
If ∼q then ∼p
Symbol ∼q → ∼p
CONNECTING ALGEBRA TO PROOF
PROPERTIES OF EQUALITY
REFELXIVE
SYMMETRIC
TRANSITIVE
ASSOCIATIVE
DISTRIBUTIVE
ADDITION
SUBTRACTION
MULTIPLICATION
DIVISION
a=a
If a b=
=
then b a
If a b=
=
and b c, =
then a c
a + (b + c) = (a + b) + c
a (b + c) = ab + ac
If a = b, then a + c = b + c
If a = b then a – c = b – c
If a = b then ab = bc
a b
=
If a b=
, then
c c
Example 1: Solve 3x= 6 −
Given: 3x= 6 −
1
x
2
Solution:
Steps
1
x
2
Reasons
1
1. 3x = 6 - x
2
2. 6=
x 12 − x
given
1. ________________
multiplication
2. ________________
addition
3. ________________
3. 7 x = 12
12
4. x =
7
division
4. ________________
YOUR TURN Supply the reasons in the following algebraic solutions:
1. Prove that if
3
x=
−9, then x =
−15.
5
3
x = −9
5
Prove: x = −15
Given:
Steps
3
x = −9
5
b. 3x = −45
c. x = −15
a.
Reasons
given
a. ____________
multiplication
b. ____________
division
c. ____________
2. Prove that if 3x − 2 =x − 8, then x =−3.
Given: 3x − 2 = x − 8
Prove: x = −3.
Steps
a. 3x − 2 = x − 8
b. 2 x − 2 =−8
Reasons
given
a. ______________
subtraction
b. ______________
c. 2 x = −6
d. x = −3
addition
c. ______________
division
d. ______________
3. Prove that if 2( x=
− 3) 8, then
=
x 7.
Given: 2( x − 3) =
8
Prove: x = 7.
Steps
a.
b.
c.
d.
2( x − 3) =
8
2x − 6 =
8
2 x = 14
x=7
Reasons
a. given
b. distribution
c. addition
d. division
4. Prove that if 3x − 4=
Given: 3x − 4=
Prove: x = 4
1
x + 6, then x= 4.
2
1
x+6
2
Steps
a. 3 x − 4=
1
x+6
2
5
x−4=
6
2
5
c. x = 10
2
d. x = 4
b.
Reasons
a. given
b. subtraction
c. addition
d. multiplication
Using Logic to Solve Problems
Maria, Jennifer, and Zi are each painting a different picture for their school’s art show. One girl is
painting a landscape, one is painting a still life, and the other is painting a portrait. Each is using a
different medium to paint the picture: oils, watercolors, and acrylics. Maria is painting a still life. The
portrait will not be painted with oils. The girl painting the landscape is using oils. Zi is using water
colors to paint. The oldest girl likes spinach. Which painting is each girl doing and what medium are
they using to paint?