Conditional Statements and Logic 2.2 Ms. Verdino.

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Presentation transcript:

Conditional Statements and Logic 2.2 Ms. Verdino

Conditional Statement A conditional statement is a statement that can be written in if-then form. “If _____________, then ______________.” If the batteries are dead, then then the TV remote won't work.

Conditional Statements have two parts: The hypothesis is the part of a conditional statement that follows “if” (when written in if-then form.) The hypothesis is the given information, or the condition. The conclusion is the part of an if-then statement that follows “then” (when written in if-then form.) The conclusion is the result of the given information.

Symbolic Logic Symbols can be used to modify or connect statements. Symbols for Hypothesis and Conclusion: Hypothesis is represented by “p”. Conclusion is represented by “q”. if p, then q or p implies q

Symbolic Logic if p, then q or p implies q is used to represent p  q Example: p: a number is prime q: a number has exactly two divisors If a number is prime, then it has exactly two divisors. p  q:

Writing A Conditional Statement Conditional statements can be written in “if-then” form to emphasize which part is the hypothesis and which is the conclusion. Hint: Turn the subject into the hypothesis. Example 1:Vertical angles are congruent.can be written as... If two angles are vertical, then they are congruent. Conditional Statement: Example 2: Seals swim. can be written as... Conditional Statement: If an animal is a seal, then it swims.

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

Conditional statements can be true or false. A conditional statement is false only when the hypothesis is true, but the conclusion is false. The truth value of a conditional statement is either true or false. If you live in Tennessee, then you live in Nashville.Statement: Counterexample:I live in Tennessee, BUT I live in Morristown. Is there a counterexample? Therefore (  ) the statement is false. Yes !!!

Is the conditional true or false? 1)If a month has 28 days, then it is February. 2)If two angles from a linear pair, then they are supplementary. 3)If a number is divisible by 5, then it is odd.

Symbolic Logic is used to represent the word “not” ~ Example 1:p: the angle is obtuse The angle is not obtuse ~p means that the angle could be acute, right, or straight. ~p: Note: Example 2:p: I am not happy ~p: I am happy ~p took the “not” out- it would have been a double negative (not not)

Symbolic Logic is used to represent the word “therefore”  Example: Therefore, the statement is false.  the statement is false

Forms of Conditional Statements Converse: Switch the hypothesis and conclusion (q  p) p  q If two angles are vertical, then they are congruent. q  p If two angles are congruent, then they are vertical.

Forms of Conditional Statements Inverse: State the opposite of both the hypothesis and conclusion. (~p  ~q) p  q : If two angles are vertical, then they are congruent. ~p  ~q: If two angles are not vertical, then they are not congruent.

Forms of Conditional Statements Contrapositive: Switch the hypothesis and conclusion and state their opposites. (~ q  ~p) p  q : If two angles are vertical, then they are congruent. ~q  ~p: If two angles are not congruent, then they are not vertical.

Forms of Conditional Statements Contrapositives are logically equivalent to the original conditional statement. If p  q is true, then  q  p is true. If p  q is false, then  q  p is false.

Compound statement A compound statement combines two or more statements.

Symbolic Logic is used to represent the word “or”  Example: p: a number is even q: a number is divisible by 3 p  q: A number is even or it is divisible by 3. i.e. 2,3,4,6,8,9,10,12,14,15,...

Symbolic Logic is used to represent the word “and”  Example: p: a number is even q: a number is divisible by 3 A number is even and it is divisible by 3. i.e. 6,12,18,24,30,36,42... p  q:

Constructing a compound statement s  j s  j s  (j  t) (s  j)  t s: We will go to the beach j: We will go out to dinner t: We will go to the movies

Truth tables

Creating a truth table Examplepqp q p  qp  q p: Ohio is a state q: There are 50 states p: Georgia is a state q: Miami is a state p: = 5 q: 2(2)= 4

Biconditional Statements When a conditional statement and its converse are both true, the two statements may be combined. Use the phrase if and only if (sometimes abbreviated: iff) Statement: If an angle is right then it has a measure of 90 . Converse: If an angle measures 90 , then it is a right angle. Biconditional: An angle is right if and only if it measures 90 .