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G.1ab Logic Conditional Statements Modified by Lisa Palen
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Conditional Statement
Defn. A conditional statement is a statement that can be written as an if-then statement. That is, as “If _____________, then ______________.”
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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.
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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.
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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
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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.
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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.
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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.
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Example 3: can be written as...
Babies are illogical. can be written as... If a person is a baby, then the person is illogical.
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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.
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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.
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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.
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Symbolic Logic Symbols can be used to modify or connect statements.
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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
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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
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is used to represent the words
is used to represent the words “if … then” or “implies”
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p q means if p, then q or p implies q
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pq: Example p: a number is prime q: a number has exactly two divisors
If a number is prime, then it has exactly two divisors.
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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.
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~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.
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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)
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is used to represent the word
is used to represent the word “and”
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pq: A number is even and it is divisible by 3.
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...
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is used to represent the word
is used to represent the word “or”
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pq: A number is even or it is divisible by 3.
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,...
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is used to represent the word
is used to represent the word “therefore”
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Example Therefore, the statement is false. the statement is false
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Different Forms of Conditional Statements
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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. Are these statements true or false? Continued…..
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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?
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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?
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If pq is true, If pq is 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.
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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
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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.
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Hypothesis iff Conclusion Hypothesis Conclusion
Biconditional A biconditional is in the form: Hypothesis if and only if Conclusion. or Hypothesis iff Conclusion Hypothesis Conclusion
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Biconditionals in symbols
Since p q means pq AND qp, p q Is equivalent to (pq) (qp)
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