Lesson 2-1 Conditional Statements
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 prime Leave 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 Example p: 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 divisors p q: 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 p q: 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 p q: 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) p q If two angles are vertical, then they are congruent. q p 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) p q : 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) p q : 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 p q is true, then q p is true. If p q is false, then q p 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: p q If an angle is right then it measures 90 . Converse: q p If an angle measures 90 , then it is right. Biconditional: p q An angle is right iff it measures 90 .
Biconditional A biconditional is in the form: Hypothesis if and only if Conclusion. or Hypothesis iff Conclusion or Hypothesis Conclusion
Biconditionals in symbols Since p q means p q AND q p, p q Is equivalent to (p q) ( q p)