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Conditional Statements. Geometry Chapter 2, Section 1. Notes. Conditional Statement: is a logical statement with two parts, a hypothesis and a conclusion. Hypothesis: are the conditions that we’re considering Conclusion: is what follows as a result of the conditions in the hypothesis.
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Conditional Statements Geometry Chapter 2, Section 1
Notes • Conditional Statement:is a logical statement with two parts, a hypothesis and a conclusion. • Hypothesis: are the conditions that we’re considering • Conclusion: is what follows as a result of the conditions in the hypothesis. • If-then form: a style of stating a conditional statement where the hypothesis comes immediately after the word if and the conclusion comes immediately after the word then. • That is, if the “if” part is satisfied then the “then” part must follow.
Notes • Example: • If-then form of a conditional statement: If it is raining outside, then the ground is wet. • Hypothesis– it is raining outside • Conclusion– the ground is wet.
Notes On your own: • Identify the hypothesis and the conclusion, then write the following conditional statement in if-then form. • A number divisible by 9 is divisible by 3 • h: a number is divisible by 9 • c: it is divisible by 3 • If a number is divisible by 9, then it is divisible by 3 • The 49ers will play in the Super Bowl XLII, if they win their next game. • h: they win their next game • c: 49ers will play in the Super Bowl XLII • If they win their next game then the49ers will play in the Super Bowl XLII
Notes • For a conditional statement to be true, it must be proven true for all cases that satisfy the conditions of the hypothesis • A single counterexample is enough to prove a conditional statement false • On Your Own: • Write a counterexample to show that the following statement is false. • If x2 = 16, then x = 4 • Counterexample: if x = -4 then x2= 16, i.e. the hypothesis is satisfied, but x does not equal 4 • This proves the statement false.
Notes • Related Conditionals: other statements formed by changing the original statement. • Converse: of a statement is formed by switching the conclusion and the hypothesis. The converse of a statement is not always true! • Example: • Original: If it’s raining outside, then the ground is wet. • Converse: If the ground is wet, then it is raining outside. • Q: is the converse true or false? • False
Notes • On Your Own: Write the converse of the following statement • Original: If a number is divisible by 9 then it is divisible by 3 • Converse: If it is divisible by 3, then a number is divisible by 9 • Q: is the converse true or false? • False • Original: If two segments are congruent, then they have the same length. • Converse: If two segments have the same length, then they are congruent. • Q: is the converse true or false? • True
Notes • Inverse: formed by negating the hypothesis and conclusion of the statement • Example • Statement: If it’s raining outside, then the ground is wet. • Inverse: If it’s not raining outside, then the ground is not wet. • On Your Own: write the inverse of the following statement • Statement: If two segments are congruent, then they have the same length. • Inverse: If two segments are not congruent, then they do not have the same length.
Notes • Contrapositive: (Combination of converse and inverse) formed by switching the hypothesis and the conclusion and negating them. • Example • Statement: If it’s raining outside, then the ground is wet. • Contrapositive: if the ground is not wet, then it is not raining outside • On Your Own • Statement: If two segments are congruent, then they have the same length. • Contrapositive: If two segments do not have the same length, then they are not congruent.
Notes • Logically Equivalent Statements: Statements that have the same truth value (i.e. when one is true, so is the other) • A statement and its contrapositive are equivalent statements • Original: If it’s raining outside, the ground is wet. • Contrapositive: If the ground isn’t wet, does that mean it isn’t raining? • Yes • Lets think about this, are these two saying the same thing? • The converse and inverse are also logically equivalent.
Conditional Statement Activity • Come up with your own conditional statement in if-then form • Write the converse, inverse, and contrapositive. • Judge the validity of all four statements. • Do the equivalent statements match up as they should and make sense? • Write counterexamples for the statements you think are false. • Be sure to: • Label the four statements, • Indicate whether each is true or false, and • Show which statements are equivalent to each other.
