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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
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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.