Section 2.2 – Good Definitions and Biconditionals

# Section 2.2 – Good Definitions and Biconditionals

## Section 2.2 – Good Definitions and Biconditionals

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1. Section 2.2 – Good Definitions and Biconditionals

2. Are the following definitions good definitions? A dog is an animal with a tail. No Wednesday is the midpoint of the week. Yes February is a month during the year. No

3. How do we know if something is a good definition? A sentence is a good definition if when written in “If, then” format both the conditional and the converse are true. Ex. Wednesday is the midpoint of the week. Conditional: If it is Wednesday then it is the midpoint of the week. (True) Converse: If it is the midpoint of the week, then it is Wednesday. (True) Since both the conditional and the converse are true this is a good definition.

4. An example of a bad definition Ex. A dog is an animal with a tail. Conditional: If it is a dog, then it is an animal with a tail. (True) Converse: If it is an animal with a tail, then it is a dog. (False) Since only the conditional is true and the converse is false, this is a bad definition.

5. Biconditionals - Making good definitions easier to write. Writing a conditional & a converse can be a lot of work, so a way was developed to simplify things. If the conditional and converse are both true, they can be combined into one sentence called a biconditional. The words “if and only if” are always used in a biconditional statement.

6. Example of a Biconditional Ex. Wednesday is the midpoint of the week. Conditional:Ifit is Wednesday, thenit is the midpoint of the week. (True) Converse:Ifit is the midpoint of the week, thenit is Wednesday. (True) Biconditional:It is Wednesday if and only if it is the midpoint of the week.

7. Your turn to try one. Water is called H2O. Conditional:Ifit is Water, thenit is H2O. Converse:Ifit is H2O, thenit is Water. Is the above a good definition? Why? Since it is a good definition, write it as a biconditional. It is water if and only if it is H2O.

8. Working backwards – Write the biconditional as two sentences. It is a right angle if and only if it has 90 degrees. • Ifit is a right angle, thenit has 90 degrees. • If it has 90 degrees, thenit is a right angle.

9. Homework P 78 (1, 5, 9, 11, 15, 18, 21, 27-30, 32, 41)