Statements and Reasoning

The definition of Statements, Negations, Conjunctions and Disjunctions Statement – a set of words and symbols that collectively make

a claim that can be classified as true or false. A statement can be given a variable name, such as P or Q.

Examples:P: 7 < 3 Q: 4 + 3 = 7

FalseTrue

Negation – a claim of the opposite of a statement. If P is a statement, then ~P is the negation.

Examples: P: It rains in Seattle. ~P: It does not rain in Seattle. Q: 4 + 3 = 7 ~Q: 4+ 3 ≠7

Opinions, Questions, and Commands are not statements, such as “Country music sucks.” or “Who is it?” or “Look out!”

Statements and Negations

Conjunction - The combination of two statements with the “AND” condition. A conjunction is only true If BOTH statements are true.

Disjunction – the combination of two statements with the “OR” condition. A disjunction can be true if EITHER statement is true.

Examples:P: It rains in Seattle Q: 4 + 3 = 7

~P and Q _______

~P or Q ________true

false

Reasoning

Intuition is often called “insight.” Conclusions are based upon a hunch. The conclusion is as often wrong as right.EXAMPLE: Before the opening kickoff of the first game of the season, Bill

predicts that his team will win the game.Induction is the type of reasoning in which specific observations lead to a

general conclusion. Induction is used in laboratory experimentation and clinical observation. Conclusions are often accurate in that they are the result of many observations.

EXAMPLE: After examination and diagnosis of several patients, a pediatrician concludes that there is an influenza epidemic in that area.

Deduction is the type of reasoning in which a general principle leads to a specific conclusion. Deduction, which is used often in geometry, is a certain method when properly applied.EXAMPLE: If an integer is even, then it is divisible by two. Since 14 is an even integer, it is divisible by two.

Conditional Statements and Their Converses

In mathematics, you will come across many _______________.

For Example: If a number is even,

then it is divisible by two.

If – then statements join two statements based on a condition:

A number is divisible by two only if the number is even.

Therefore, if – then statements are also called __________ __________ .

if-then statements

conditional statements

You will learn to write statements in if-then form and writethe converse of the statements.

Conditional Statements and Their Converses

Conditional statements have two parts.

The part following if is the _________ .premise

The part following then is the _________ .conclusion

If a number is even, then the number is divisible by two.

a number is even

the number is divisible by two.

Premise:

Conclusion:

Conditional Statements and Their Converses

EXAMPLE: The following valid argument is based upon the use of the Law of Detachment.(1) If a person lives in Toronto, then he/she lives in Canada.(2) Mary Francis lives in Toronto.(C) Mary Francis lives in Canada.

Law of Detachment1) If P, then Q.2) P---------------------(C) Q

means “therefore”

A common mistake in applying the Law of Detachment is called“asserting the conclusion.” The following argument is NOT valid.

(1) If a person lives in Toronto, then he/she lives in Canada.

(2) Mary Francis lives in Canada.

(C) Mary Francis lives in Toronto.

Conditional Statements and Their Converses

How do you determine whether a conditional statement is true or false?

ConditionalStatement

True orFalse

Why?

If it is the 4th of July(in the U.S.), then it is aholiday.

True The statement is true becausethe conclusion follows fromthe premise.

If an animal lives in the water, then it is a fish.

False You can show that the statement is false by givingone counterexample.

Whales live in water, butwhales are mammals, not fish.

Conditional Statements and Their Converses

There are different ways to express a conditional statement.The following statements all have the same meaning.

If you are a member of Congress, then you are a U.S. citizen.

All members of Congress are U.S. citizens.

You are a U.S. citizen if you are a member of Congress.

You write two other forms of this statement: “If two lines are parallel, then they never intersect.”

All parallel lines never intersect.

Lines never intersect if they are parallel.

Possible answers:

Conditional Statements and Their Converses

The ________ of a conditional statement is formed by exchanging the premise and the conclusion.

converse

Conditional: If a figure is a triangle, then it has three angles.a figure is a triangle it has three angles

NOTE: You often have to change the wording slightly so that the converse reads smoothly.

Converse: If the figure has three angles, then it is a triangle.

a figure is a triangleit has three anglesConverse: If _______________, then ________________.

Conditional Statements and Their Converses

Write the converse of the following statements.State whether the converse is TRUE or FALSE.If FALSE, give a counterexample:

“If you are at least 16 years old, then you can get a driver’s license.”

If ________________________, then _______________________.you can get a driver’s license you are at least 16 years old

“If today is Saturday, then there is no school.

If _______________, then ______________.there is no school today is Saturday

TRUE!

FALSE!

We don’t have school on New Years day which may fall on a Monday.

Example: Ten Best FriendsYou could have a set made up of your ten best friends:•{alex, blair, casey, drew, erin, francis, glen, hunter, ira, jade} Each friend is an "element" (or "member") of the set (it is normal to use lowercase letters for them.)  Now let's say that alex, casey, drew and hunter play Soccer:Soccer = {alex, casey, drew, hunter}(The Set "Soccer" is made up of the elements alex, casey, drew and hunter).                 And casey, drew and jade play Tennis:Tennis = {casey, drew, jade}You could put their names in two separate circles:

Sets and Venn Diagrams

UnionYou can now list your friends that play Soccer OR Tennis. This is called a "Union" of sets and has the special symbol ∪:Soccer ∪ Tennis = {alex, casey, drew, hunter, jade}Not everyone is in that set ... only your friends that play Soccer or Tennis.We can also put it in a "Venn Diagram": Venn Diagram: Union of 2 SetsA Venn Diagram is clever because it shows lots of information:•Do you see that alex, casey, drew and hunter are in the "Soccer" set?•And that casey, drew and jade are in the "Tennis" set?•And here is the clever thing: casey and drew are in BOTH sets!

Sets and Venn Diagrams

Intersection"Intersection" is when you are in BOTH sets.In our case that means they play both Soccer AND Tennis ... which is casey and drew.

The special symbol for Intersection is an upside down "U" like this: ∩And this is how we write it down:Soccer ∩ Tennis = {casey, drew}In a Venn Diagram: Venn Diagram: Intersection of 2 Sets

Sets and Venn Diagrams

Sets and Venn Diagrams

Sets and Venn Diagrams

Often when we are solving an equation and we get “no solution”,Another way to put it is that the solution is the empty set,

http://www.mathsisfun.com/sets/venn-diagrams.html

Sets and Venn Diagrams

Using Venn Diagrams to verify conclusions:Example 8/111.If a student plays on the Rockville High School boys’ varsity basketball team, then he is a talented athlete.2. Todd plays on the Rockville High School boys' varsity basketball team.

What can we conclude?

The statement “If P, then Q” is sometimes expressed in the form, “All P are Q” which can be illustrated like this:

P

Q

Let P = players on the Rockville boys’ varsity basketball team.Q = talented athletesT= Todd

If follows that since All P are part of Q and T is in P, then T is in Q.

●T

athlete. talenteda is Todd ),(therefore

Sets and Venn Diagrams

SubsetsWhen we define a set, if we take pieces of that set, we can form what is called a subset. So for example, we have the set {1, 2, 3, 4, 5}. A subset of this is {1, 2, 3}. Another subset is {3, 4} or even another, {1}. However, {1, 6} is not a subset, since it contains an element (6) which is not in the parent set. In general:A is a subset of B if and only if every element of A is in B.So let's use this definition in some examples. Is A a subset of B, where A = {1, 3, 4} and B = {1, 4, 3, 2}? 1 is in A, and 1 is in B as well. So far so good. 3 is in A and 3 is also in B. 4 is in A, and 4 is in B.That's all the elements of A, and every single one is in B, so we're done.Yes, A is a subset of B

Note that 2 is in B, but 2 is not in A. But remember, that doesn't matter, we only look at the elements in A.

In the Rockville basketball example, Since All Basketball Players are Talented Athletes (If P, then Q),Then we can also say

{1, 3, 4}

{1, 4, 3, 2}

More NotationWhen we say that A is a subset of B, we write BA

QP

Subsets