Logic

The study of critical thinking

Mathematical Sentence

A statement of fact…called a statement

Can be judged as true or false

Not a question, command, or phrase

Examples of sentences

Determine if the following are mathematical sentences (statements). If so, are they true or false?

An isosceles triangle has two congruent sides.

10 + 4 = 15

Did you get that one right?

All quadrilaterals

Negation

When we change the truth value of a mathematical sentence, we negate the statement, or create the negation.

The negation is usually formed by adding or removing some form of the word not.

Examples

The Word of the Week is aggrandize.

The Reds beat the Pirates on Tuesday night.

It is not going to snow in Marlboro tomorrow.

Compound Sentence

A compound statement is formed when two or more thoughts are connected in one mathematical sentence.

Examples of Compound Statements

Today is a vacation day, and I sleep late.

You can call me at 10 o’clock, or you can call me at 2 o’clock.

If you are going to the beach, then you should take the sunscreen.

AND Statements

A conjunction is a compound statement formed by combining two mathematical sentences (or facts) using the word “and.”

A conjunction is true only when BOTH statements are true.

Examples

Blue is a color, and 7 + 3 = 10.

One hour equals exactly 55 minutes, and one minute equals exactly 60 seconds.

3 + 4 = 6, and all dogs meow.

OR Statements

A disjunction is a compound statement formed by combining two mathematical sentences (or facts) using the word “or.”

A disjunction is true when EITHER or BOTH statements are true.

Examples

Blue is a color, or 7 + 3 = 10.

One hour equals exactly 55 minutes, or one minute equals exactly 60 seconds.

3 + 4 = 6, or all dogs meow.

Negation of a Conjunction

Original : 3 + 4 = 6 and all dogs meow. F F F

Negation : 3 + 4 ≠ 6 or all dogs do not meow. T T T

Your turn. Negate, “ Two points determine a line, and triangles do not have 3 sides.”

Negation of a Disjunction

Original : Blue is a color, or 3 + 7 = 9. TT F

Negation : Blue is not a color, and 3 + 7 ≠ 9. FF T

Your turn. Negate, ”Leaves fall in autumn, or snow falls in July.”

IF…THEN

A conditional is a compound statement formed by combining two mathematical sentences (or facts) using the words “if…then…”

A conditional can also be called an implication.

A conditional has two parts: the hypothesis and the conclusion.

A conditional is only false when a true hypothesis results in a false conclusion.

Hypothesis

The hypothesis is the statement that usually follows the word “if” in the conditional statement.

The hypothesis is the condition that must be met before the entire statement can be judged true or false.

Conclusion

The conclusion is the statement that usually follows the word “then” in the conditional statement.

A true conclusion does not necessarily mean a true conditional statement.

Truth Values for If…then…”

true hypothesis, true conclusion: true conditional

true hypothesis, false conclusion: false conditional

false hypothesis, true conclusion: true conditional

false hypothesis, false conclusion: true conditional

ExamplesTeacher says: “If you participate in class, then you will get extra points.”

Suppose you participate in class, and you get extra points.Is the teacher’s statement true or false?

Suppose you participate in class, and you do not get extra points.Is the teacher’s statement true or false?

Suppose you do not participate in class.Can we judge the truth of the teacher’s statement?

WHEN THE HYPOTHESIS IS FALSE, WE CANNOT JUDGE THE TRUTH OF THE CONDITIONAL, SO THE DEFAULT TRUTH VALUE IS TRUE.

Examples of the Conditional Statement

Triangles have four angles if parallel lines intersect.

1. Rewrite the statement in if…then form:

2. Identify the hypothesis.

3. Identify the conclusion.

4. What is the truth value?

Student Examples of Conditional Statements

Student Examples of Conditional Statements

Student Examples of Conditional Statements

The Converse

The converse is a rearrangement of a conditional statement.

The converse of a conditional statement is formed by interchanging the hypothesis and the conclusion of the original statement.

The truth value of the converse is independent of the truth value of the original conditional statement.

Converse Examples

Conditional: If the space shuttle was launched, then a cloud of smoke was seen.

Converse: If a cloud of smoke was seen, then the space shuttle was launched.

Suppose a cloud of smoke was seen, but no space shuttle was launched. What is the truth value of each of the above statements?

More Converse Examples

Conditional: If you grew up in Alaska, then you have seen snow.

Converse: If you have seen snow, then you grew up in Alaska.

Suppose you have seen snow, but you did not grow up in Alaska. What is the truth value of each of the above statements?

Student Examples of Converse Statements

The Inverse

The inverse is a variation of a conditional statement.

The inverse of a conditional statement is formed by negating the hypothesis and negating the conclusion of the original statement.

The truth value of the inverse is independent of the truth value of the original conditional.

Inverse Examples

Conditional: If the space shuttle was launched, then a cloud of smoke was seen.

Inverse: If the space shuttle was not launched, then a cloud of smoke was not seen.

Suppose, once again, that a cloud of smoke was seen, but no space shuttle was launched. What is the truth value of each of the above statements?

More Inverse Examples

Conditional: If you grew up in Alaska, then you have seen snow.

Inverse: If you did not grow up in Alaska, then you have not seen snow.

Suppose you have seen snow, but you did not grow up in Alaska. What is the truth value of each of the above statements?

Student Examples of Inverse Statements

Compare

Compare the truth values we just found for the converse examples with the truth values we found for the inverse examples.

What do you see?

The Contrapositive

The contrapositive is a rearrangement and variation of a conditional statement.

The contrapositive of a conditional statement is formed by negating both the hypothesis and the conclusion, and then, interchanging the resulting negations. (It does BOTH jobs of the inverse and converse.)

The truth value of the contrapositive is the same as the truth value of the original statement.

Contrapositive Examples

Conditional: If the space shuttle was launched, then a cloud of smoke was seen.

Contrapositive: If a cloud of smoke was not seen, then the space shuttle was not launched.

Suppose a cloud of smoke was seen and the space shuttle was not launched. What is the truth value of each of the above statements?

More Contrapositve Examples

Conditional: If you grew up in Alaska, then you have seen snow.

Contrapositive: If you have not seen snow, then you did not grow up in Alaska.

Suppose, once again, that you have seen snow, but you did not grow up in Alaska. What is the truth value of each of the above statements?

Student Examples of Contrapositive Statements

Compare

How did the truth values of the contrapositive statements in the last two examples compare to the truth values of the original conditional statements?

The Really Big Deal

When two statements have the same truth value, we say that they are

logically equivalent.(Equivalent means “the same”.)

Logical Equivalence

In our examples, the converse and the inverse had the same truth values.

The converse and inverse are logically equivalent.

In our examples, the conditional and the contrapositive had the same truth values.

The conditional and the contrapositive are logically equivalent.

IF AND ONLY IF

The biconditional is a compound statement formed by combining two statements using the words “if and only if.”

The biconditional is a shortened version of conjunction joining the conditional statement with its converse.

Like any other “AND” statement, the biconditional is true only when BOTH the conditional and its converse are true.

Example of the Biconditional

Conditional: If two lines in a plane intersect, then the linesare not parallel.

Converse: If two lines in a plane are not parallel, then the lines intersect.

Conjunction: If two lines in a plane intersect, then the lines are not parallel; and, if they are not parallel, then

the lines intersect.

Biconditional: Two lines in a plane intersect if and only if they are not parallel.

Biconditional: Two lines in a plane intersect if and only if they are not parallel.

-or-

Biconditional: Two lines in a plane are not parallel if and only if they intersect.

-or-

Biconditional: Two lines in a plane intersect iff they are not parallel.

Ways to Write the Biconditional

“iff” is shorthand for “if and only if”

Biconditional Examples

Conditional: If a polygon has four sides, then it is a quadrilateral. T

Converse: If a polygon is a quadrilateral, then it has four sides. T

Conjunction: If a polygon has four sides, then it is a quadrilateral; and, if a polygon is a quadrilateral, then it has four sides. T

Biconditional: A polygon is a quadrilateral if and only if it has four sides.

Practice with the Biconditional

Conditional: If you like music, then you watch music videos.

Converse:

Conjunction:

Biconditional:

The Law of Detachment

A conditional statement is true. The hypothesis of that conditional statement is true. We can conclude that the conclusion of that conditional

statement is also true.

Example: If Sandy takes music, then she will take French.

Sandy takes music.

Therefore, by the Law of Detachment, we can conclude that Sandy also takes French.

The Law of Disjunctive Inference

A disjunction is assumed to be true. One of the smaller statements is false We can conclude that the other of the smaller

statements is true.

Example: Grace goes shopping, or Grace goes to a meeting.

Grace does not go shopping.

Therefore, by the Law of Disjunctive Inference, we can conclude that Grace goes to a meeting.

The Law of Syllogism

A conditional statement is true. A second conditional, which has the

conclusion of the first as its hypothesis is also true.

We can conclude that a third conditional, having the same hypothesis as the first, and the conclusion of the second, is also true.

Example of the Law of Syllogism

If I do my homework, I will earn mastery on the regents exam.

If I earn mastery on the regents exam, I will be accepted to a good college.

Therefore, by the Law of Syllogism, we can conclude, “If I do my homework, then I will be accepted to a good college.”