Lesson 2-1 Conditional Statements. Conditional Statement Defn. A conditional statement is a statement that can be written as an if- then statement. That.

Lesson 2-1

ConditionalStatements

Conditional Statement

Defn. A conditional statement is a statement that can be written as an if-then statement. That is, as

“If _____________, then ______________.”

Example:

If your feet smell and your nose runs,

then you're built upside down.

Conditional Statements have two parts:

The hypothesis is the part of a conditional statement that follows “if” (when written in if-then form.)

It is the given information, or the condition.

If a number is prime, then a number has exactly two divisors.

Hypothesis: a number is primeLeave off “if” and comma.

Conditional Statements have two parts:

The conclusion is the part of a conditional statement that follows “then” (when written in if-then form.)

It is the result of the given information.

If a number is prime, then a number has exactly two divisors.

Conclusion: a number has exactly two divisors

Leave off “then” and period

Conditional statements can be put into an “if-then” form to clarify which part is the hypothesis and which is the conclusion.

Method: Turn the subject into a hypothesis.

Rewriting Conditional Statements

Example 1:

Vertical angles are congruent.

If two angles are vertical, then they are congruent.

can be written as...

Example 2:

Seals swim.

If an animal is a seal, then it swims.

can be written as...

Example 3:

Babies are illogical.

If a person is a baby, then the person is illogical.

can be written as...

IF …THEN vs. IMPLIES

Two angles are vertical implies they are congruent.

Another way of writing an if-then statement is using the word implies.

Conditional Statements

can be true or false:• A conditional statement is false

only when the hypothesis is true, but the conclusion is false.

• A counterexample is an example used to show that a statement is not always true and therefore false.

Counterexample

Therefore () the statement is false.

Statement: If you live in Virginia, then you live in Richmond, VA.

Is there a counterexample?

YES... Anyone who lives in Virginia, but not Richmond, VA.

Symbols can be used to modify or connect statements.

Symbolic Logic

Symbols for Hypothesis and Conclusion

if p, then q or

p implies q

Lower case letters, such as p and q, are frequently used to represent the hypothesis and conclusion.

Symbols for Hypothesis and Conclusion

if p, then q or p implies q

Examplep: a number is prime

q: a number has exactly two divisors

If a number is prime, then it has exactly two divisors.

is used to represent the words

“if … then”or

“implies”

p q

if p, then q

or

p implies q

means

Example

p: a number is prime q: a number has exactly two divisorspq: If a number is prime, then

it has exactly two divisors.

is used to represent the word

“not”

~

•~ p is the negation of p.•The negation of a statement is the denial of the statement. Add or remove the word “not.”•To negate, write ~ p.

Example

p: the angle is obtuse

~p: the angle is not obtuse

Be careful because ~p means that the angle could be acute, right, or straight.

Example

p: James doesn’t like fish.

~p: James likes fish.

Notice: ~p took the “not” out… it would have been a double negative (not not)

is used to represent the word

“and”

Example

p: a number is even q: a number is divisible by 3

pq: A number is even and it is divisible by 3.

6,12,18,24,30,36,42...

is used to represent the word

“or”

Example

p: a number is even q: a number is divisible by 3

pq: A number is even or it is divisible by 3.

2,3,4,6,8,9,10,12,14,15,...

is used to represent the word

“therefore”

Example

Therefore, the statement is false.

the statement is false

Different Forms of Conditional Statements

Forms of Conditional Statements

Converse: Statement formed from a conditional statement by switching the hypothesis and conclusion (q p)

pq If two angles are vertical, then they are congruent.

qp If two angles are congruent, then they are vertical.

Continued…..Are these statements true or false?

Forms of Conditional Statements

Inverse: Statement formed from a conditional statement by negating both the hypothesis and conclusion.

(~p~q)

pq : If two angles are vertical, then they are congruent.

~p~q: If two angles are not vertical, then they are not congruent.

Are these statements true or false?

Forms of Conditional Statements

Contrapositive: Statement formed from a conditional statement by switching and negating both the hypothesis and conclusion.

(~q~p)

pq : If two angles are vertical, then they are congruent.

~q~p: If they are not congruent, then two angles are not vertical

Are these statements true or false?

Contrapositives are logically equivalent to the original conditional statement.

• If pq is true, then qp is true.

• If pq is false, then qp is false.

Biconditional • When a conditional statement and its

converse are both true, the two statements may be combined.

• A statement combining a conditional statement and its converse is a biconditional.

• Use the phrase if and only if which is abbreviated iff

• Use the symbol

Definitions are always biconditional

Statement: pq If an angle is right then it measures 90.Converse: qp If an angle measures 90, then it is right.Biconditional: pq An angle is right iff it measures 90.

Biconditional • A biconditional is in the form:Hypothesis if and only if Conclusion.

orHypothesis iff Conclusion

or

Hypothesis Conclusion

Biconditionals in symbols

Since p q means pq AND qp,

p q Is equivalent to(pq) (qp)