G.1ab Logic Conditional Statements Modified by Lisa Palen

Conditional Statement Defn. A conditional statement is a statement that can be written as an if-then statement. That is, as “If _____________, then ______________.”

If your feet smell and your nose runs, then you're built upside down. 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

Rewriting Conditional Statements 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.

Example 1: can be written as... If two angles are vertical, Vertical angles are congruent. can be written as... If two angles are vertical, then they are congruent.

Example 2: can be written as... If an animal is a seal, then it swims. Seals swim. can be written as... If an animal is a seal, then it swims.

Example 3: can be written as... Babies are illogical. can be written as... If a person is a baby, then the person is illogical.

Two angles are vertical implies they are congruent. IF …THEN vs. IMPLIES Another way of writing an if-then statement is using the word implies. Two angles are vertical implies 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. A counterexample is an example used to show that a statement is not always true and therefore false.

Counterexample Statement: If you live in Virginia, then you live in Richmond, VA. Is there a counterexample? Anyone who lives in Virginia, but not Richmond, VA. YES... Therefore () the statement is false.

Symbolic Logic Symbols can be used to modify or connect statements.

Symbols for Hypothesis and Conclusion Lower case letters, such as p and q, are frequently used to represent the hypothesis and conclusion. if p, then q or p implies q

Symbols for Hypothesis and Conclusion Example p: a number is prime q: a number has exactly two divisors If a number is prime, then it has exactly two divisors. if p, then q or p implies q

is used to represent the words  is used to represent the words “if … then” or “implies”

p  q means if p, then q or p implies q

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.

is used to represent the word ~ 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.

~p: the angle is not obtuse 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  is used to represent the word “and”

pq: A number is even and it is divisible by 3. 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  is used to represent the word “or”

pq: A number is even or it is divisible by 3. 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  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. Are these statements true or false? Continued…..

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?

If pq is true, If pq is 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.

Hypothesis iff Conclusion Hypothesis  Conclusion Biconditional  A biconditional is in the form: Hypothesis if and only if Conclusion. or Hypothesis iff Conclusion Hypothesis  Conclusion

Biconditionals in symbols Since p  q means pq AND qp, p  q Is equivalent to (pq)  (qp)