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Lesson 2.2

Lesson 2.2. Conditional Statements. Conditional Statement. A conditional statement is a logical statement that can be written in if-then form. “ If _____________, then ______________.”. Definition:. If your feet smell and your nose runs , then you're built upside down .

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Lesson 2.2

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  1. Lesson 2.2 Conditional Statements

  2. Conditional Statement A conditional statement is a logical statement that can be written in if-then form. “If _____________, then ______________.” Definition: Ifyour feet smell and your nose runs, then you're built upside down. Example:

  3. Conditional Statement - continued 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 conclusionis 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.

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

  5. If …Thenvs. Implies Two angles are verticalimplies 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.

  6. 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. Statement: If you live in Virginia, then you live in Richmond. Is there a counterexample? Yes !!! Counterexample: I live in Virginia, BUT I live in Leesburg. Therefore () the statement is false.

  7. 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

  8. Symbolic Logic - continued if p, then q or p implies q p  q is used to represent 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.

  9. is used to represent the word Symbolic Logic - continued ~ “not” Example 1: p: the angle is obtuse ~p: The angle is not obtuse Note: ~p means that the angle could be acute, right, or straight. 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)

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

  11. is used to represent the word Symbolic Logic-continued  “or” Example: p: a number is even q: a number is divisible by 3 pq: A number is even orit is divisible by 3. i.e.2, 3, 4, 6, 8, 9, 10, 12, 14,1 5, ...

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

  13. Forms of Conditional Statements Converse:Switch the hypothesis and conclusion (p  q becomes q  p) Conditional: pqIftwo angles are vertical, thenthey are congruent. Converse: qpIftwo angles are congruent, thenthey are vertical.

  14. Forms of Conditional Statements Inverse:State the opposite(negation) of both the hypothesis and conclusion. (p  q becomes ~p~q) Conditional: pq :Iftwo angles are vertical, thenthey are congruent. Inverse: ~p~q:Iftwo angles are not vertical, thenthey are not congruent.

  15. Forms of Conditional Statements Contrapositive: Switch the hypothesis and conclusion and statethe opposites(negations). (p  q becomes ~q~p) Conditional: pq : Iftwo angles are vertical, thenthey are congruent. Contrapositive: ~q~p:Iftwo angles are not congruent, thenthey are not vertical.

  16. 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.

  17. Biconditional • 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. (true) Converse: If an angle measures 90, then it is a right angle. (true) Since both the original (conditional) statement and the converse are true, we can write the biconditional: Biconditional:An angle is a right angle if and only if it measures 90.

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