Deductive and Inductive reasoning

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Presentation transcript:

Deductive and Inductive reasoning Ch 2

Examples of deductive reasoning

Examples of inductive reasoning

Do you agree with the following conditional? If something is an obligation, then it is really an opportunity. Can you name an obligation that is not actually a chance for you to benefit? In other words, you may feel obligated to come home before curfew, but what opportunity will you miss if you choose to stay out late? If you choose to skip homework, what opportunity are you missing. If you don’t see the benefit of your obligations, why are you obligated, why not just ignore that obligation?

Conditional statements In order to understand a logical argument it helps to set it up as a conditional statement. A Conditional is an if-then statement If he is from Los Angeles, then he is from California If it is not plugged in, then it will not work The “if” part of the statement is called the hypothesis The “then”part of the statement is called the conclusion

State whether the conclusion is valid If a figure has 3 sides, then it is a triangle.

State whether the conclusion is valid If two angles share the same vertex, then they are adjacent.

State whether the conclusion is valid If two angles are adjacent, then they are congruent.

Try these by yourself, identify the hypothesis and conclusion and decide if the conditional is true

Rewrite as a conditional statement

Write a conditional statement that each Venn diagram illustrates

Counterexamples A conditional can quickly be proven false by finding a counterexample. A counterexample is any example that disproves the conditional If a number is even, then it is not prime. “2 is prime and even” is a counterexample that disproves the above statement. For a conditional to be true, the conclusion must be true any and every time the hypothesis is true.

Converse The converse of a conditional is the statement you get when you exchange the hypothesis and the conclusion For example, given a conditional of “if an angle is less than 90 degrees it is acute” the converse would be, “If an angle is acute, then it is less than 90 degrees” Converse statements are not always true, is the converse of the following true? Why or why not? “If an angle is 15 degrees then it is acute.”

Inverse We can get the inverse of a conditional by negating (making negative) the hypothesis and conclusion. For example, given a conditional of “if an angle is less than 90 degrees then it is acute” the inverse would be, “If an angle is not less than 90 degrees, then the angle is not acute” Notice we just changed “is” to “is not.” Inverse statements are not always true, is the inverse of the following true? Why or why not? “If an angle is 15 degrees then it is acute.”

Contrapositives We can get the contrapositive of a statement by finding the converse of a conditional and then negating both the hypothesis and conclusion. For example, given a conditional of “if an angle is less than 90 degrees then it is acute” the contrapositive would be, “If an angle is not acute, then it is not less than 90 degrees” Contrapositives are not always true, is the contrapositive of the following true? Why or why not? “If an angle is 15 degrees then it is acute.”

Equivalent statements Conditionals and contrapositives are equivalent statements. This means that they are either both true or both false. In other words, if a conditional is true, its contrapositive is true too and if the conditional is false, its contrapositive is also false. Inverse and converse are also equivalent statements. None of this means that the converse and inverse are never true, just not in this case. The point here is their relationship as equivalent.

Find the converse, inverse and contrapositive and determine the truth of each Conditional- If you are a quarterback, you play football. Converse Inverse Contrapositive

Truth Value A conditional can be evaluated as true or false, this is called its Truth Value A conditional is true if you can show that every time the hypothesis is true, the conclusion is also true. Remember that just one counterexample can show the conditional false

Notation We use notation for conditionals, p stands for the hypothesis and q stands for the conqlusion. We can write p q which means “if p, then q” We will use this to discuss the other statements related to conditionals

Table of related conditionals

Try these

Try these