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Conditional Statements

Lesson 2-1. Conditional Statements. Conditional Statements have two parts:. Hypothesis ( denoted by p ) and Conclusion ( denoted by q ). Hypothesis (p). Phrase following “if” the given information. Conclusion (q). Phrase following “then” the result of the given information.

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Conditional Statements

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  1. Lesson 2-1 Conditional Statements

  2. Conditional Statements have two parts: • Hypothesis (denoted byp) and • Conclusion (denoted byq)

  3. Hypothesis (p) • Phrase following “if” • thegiveninformation

  4. Conclusion (q) • Phrase following “then” • theresult of the giveninformation

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

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

  7. If two angles are vertical, then they are congruent. p implies q Hypothesis (p): two angles are vertical Conclusion (q): they are congruent

  8. 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.

  9. Giving a Counterexample Statement: If you live in Virginia, then you live in Richmond. Therefore () the statement is false. True or False? Give a counterexample: Henry lives in Virginia, BUT he lives in Ashland.

  10. Symbolic Logic

  11. Symbols can be used to modify or connect statements.

  12. is used to represent the word • “therefore”

  13. Example H : I watch football H Therefore,I watch football

  14. is used to represent • implies • used in if … then

  15. 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.

  16. ~ is used to represent the word • “not” or • “negate”

  17. Example w: the angle is obtuse ~w:the angle is not obtuse Be careful because ~w means that the angle could be acute, right, or straight

  18. Example r: I am not happy ~r:I am happy Notice: ~r took the “not” out… it would have been a double negative (not not)

  19. is used to represent the word • “and”

  20. Example a:a number is even b: a number is divisible by 3 ab:A number is even and it is divisible by 3. 6,12,18,24,30,36,42...

  21.  is used to represent the word • “or”

  22. Example a: a number is even b: a number is divisible by 3 ab:A number is even or it is divisible by 3. 2,3,4,6,8,9,10,12,14,15,...

  23. iffis used to represent the phrase • “if and only if”

  24. Example h: I watch football k: the Eagles are playing h iff k I watch football if and only if the Eagles are playing

  25. Different Forms of Conditional Statements

  26. A conditional statement can be written in three different ways. These three new conditional statements can be true or false. EXAMPLE: pq If two angles are vertical, then they are congruent.

  27. Converse:q  p SWITCH (p and q but not if or then) Iftwo angles are congruent, thenthey are vertical.

  28. Inverse:~p~q NEGATION (keep same order) Iftwo angles are not vertical, thenthey are not congruent.

  29. Contrapositive:~q~p SWITCH and NEGATE Iftwo angles are not congruent, thenthey are not vertical.

  30. 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.

  31. Biconditional • If a conditional statement and its converse are both true, then the two statements may be combined. • using the phrase if and only if (iff)

  32. Definitions are always biconditional • 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 iff it measures 90

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