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This lesson explores conditional statements, which consist of a hypothesis and a conclusion, specifically in if-then form. We will cover how to identify these components and how to rewrite statements into conditional format. The distinctions between true and false conditional statements will be examined, along with the concepts of negation, converse, inverse, and contrapositive. Examples will highlight the practical application of these concepts. This lesson is essential for grasping logical reasoning in mathematics.
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Notes on Logic Analyze Conditional Statements Lesson 4.3 Page 204
Definitions A conditional statement is a logical statement that has two parts, a hypothesis and a conclusion. When a conditional statement is written in if-then form, the “if” part contains the hypothesis and the “then” part contains the conclusion. Symbol form: “If p, then q.” p implies q (p→q)
Example: If the measure of angle A is 30⁰, then angle A is acute. Hypothesis: the measure of angle A is 30⁰ (does not include the word if) Conclusion: angle A is acute (does not include the word then)
Rewriting Statements You need to know how to rewrite statements into conditional statements. • All mammals breathe oxygen. If an animal is a mammal, then it breathes oxygen. • Two points are collinear if they lie on the same line. If two points lie on the same line, then they are collinear. • A number divisible by 9 is also divisible by 3. If a number is divisible by 9, then it is also divisible by 3.
Conditional Statements *Conditional statements can be true or false. *If it is true, then it is true for all cases of the conclusion. *If it is false, then you can find one example where the hypothesis is met but the conclusion is not. (This is called a counterexample).
Examples: True or False? • If a red car drives by, then it is a Mustang. False Counterexample – a red Honda Civic drove by • If angle B is obtuse, then the measure of angle B is 102⁰. False Counterexample – the measure of angle B is 100⁰ • If the measure of angle A is 30⁰, then angle A is acute. True
Definitions The negation of a statement is the opposite of the original statement. (the negation of p is ~p) Example: the measure of angle A = 30⁰ Negation: the measure of angle A ≠ 30⁰
To write the converse of a conditional statement, switch the hypothesis and conclusion. (q→p) Example: Conditional Statement: If the measure of angle A is 30⁰, then angle A is acute. Converse: If angle A is acute, then the measure of angle A is 30⁰.
To write the inverse of a conditional statement, negate both the hypothesis and conclusion (keeping the order of the sentence the same). (~p→~q) Example: Conditional Statement: If two segments are congruent, then their measures are equal. Inverse: If two segments are not congruent, then their measures are not equal.
To write the contrapositive of a conditional statement, first write the converse and then negate both the hypothesis and conclusion. (~q → ~p) Example: Conditional Statement: If two segments are congruent, then their measures are equal. Contrapositive: If the measures of two segments are not equal, then the two segments are not congruent.
Write the converse, inverse, and contrapositive of the following conditional statement. If an animal is a fish, then it can swim. Converse (q→p): If an animal can swim, then it is a fish. Inverse (~p→~q): If an animal is not a fish, then it cannot swim. Contrapositive (~q → ~p): If an animal cannot swim, then it is not a fish.
