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Conditional Statements: Definition, Symbolic Logic, and Forms

This lesson covers the definition of conditional statements, their symbolic logic representation, and different forms such as converse, inverse, contrapositive, and biconditional.

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Conditional Statements: Definition, Symbolic Logic, and Forms

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  1. Conditional Statements Lesson 2-1 Lesson 2-1 Conditional Statements

  2. Conditional Statement Definition: A conditional statement is a statement that can be written in if-then form. “If _____________, then ______________.” Example: Ifyour feet smell and your nose runs, then you're built upside down. Continued…… Lesson 2-1 Conditional Statements

  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. Lesson 2-1 Conditional Statements

  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. Lesson 2-1 Conditional Statements

  5. If …Then vs. 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. Lesson 2-1 Conditional Statements

  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. Yes !!! Is there a counterexample? Counterexample: I live in Virginia, BUT I live in Glen Allen. Therefore () the statement is false. Lesson 2-1 Conditional Statements

  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 Continued….. Lesson 2-1 Conditional Statements

  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. Continued….. Lesson 2-1 Conditional Statements

  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) Lesson 2-1 Conditional Statements

  10. is used to represent the word Symbolic Logic - continued  “and” p:a number is even q: a number is divisible by 3 Example: A number is even and it is divisible by 3. i.e. 6,12,18,24,30,36,42... pq: Lesson 2-1 Conditional Statements

  11. is used to represent the word Symbolic Logic-continued  “or” p: a number is even q: a number is divisible by 3 Example: pq: A number is even or it is divisible by 3. i.e.2,3,4,6,8,9,10,12,14,15,... Lesson 2-1 Conditional Statements

  12. is used to represent the word Symbolic Logic - continued  “therefore” Example: Therefore, the statement is false.  the statement is false Lesson 2-1 Conditional Statements

  13. Forms of Conditional Statements Converse: Switch the hypothesis and conclusion (q  p) pqIftwo angles are vertical, thenthey are congruent. qpIftwo angles are congruent, thenthey are vertical. Continued….. Lesson 2-1 Conditional Statements

  14. Forms of Conditional Statements Inverse:State the opposite of both the hypothesis and conclusion. (~p~q) pq :Iftwo angles are vertical, thenthey are congruent. ~p~q:Iftwo angles are not vertical, thenthey are not congruent. Lesson 2-1 Conditional Statements

  15. Forms of Conditional Statements Contrapositive: Switch the hypothesis and conclusion and state their opposites. (~q~p) pq : Iftwo angles are vertical, thenthey are congruent. ~q~p:Iftwo angles are not congruent, thenthey are not vertical. Lesson 2-1 Conditional Statements

  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. Lesson 2-1 Conditional Statements

  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. Converse:If an angle measures 90, then it is a right angle. Biconditional:An angle is right if and only if it measures 90. Lesson 2-1 Conditional Statements

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