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# Mathematical Reasoning

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### Mathematical Reasoning

1. 1. Statement A statement is a mathematical sentence which can be judged to be true or false
2. 2. Example <ul><li>“Garfield is a cartoon character.&quot; </li></ul><ul><li> a true statement. </li></ul><ul><li>&quot;A pentagon has exactly 4 sides.&quot; </li></ul><ul><li>is a false statement. </li></ul><ul><li>&quot;Did you get that one right?&quot; </li></ul><ul><li>Not a statement </li></ul><ul><li>“Stop the bus.” </li></ul><ul><li>Not a statement </li></ul>
3. 3. Quantifiers <ul><li>Words that indicate number of objects or cases referred to in a statement. </li></ul><ul><li>Example : All, Every, Any </li></ul><ul><li> Some, Several, One of, part of </li></ul>
4. 4. Examples – True Statements <ul><li>All squares have 4 sides </li></ul><ul><li>Some kites have one axis of symmetry </li></ul><ul><li>All cows have 4 legs. </li></ul>
5. 5. Examples – False Statement <ul><li>All girls like pink. </li></ul><ul><li>All fractions are smaller than one </li></ul>
6. 6. Construct a true statement using the quantifier ‘all’ or ‘some’.
7. 7. Negation <ul><li>a negation of a statement is formed by placing the  word &quot;not&quot; or no into the original statement.  </li></ul><ul><li>The negation will always have the opposite truth value of the original statement.   </li></ul><ul><li>Under negation, what was TRUE, will become FALSE - or - what was FALSE, will become TRUE </li></ul>
8. 8. <ul><li>Examples: </li></ul><ul><li>1. Original Statement:  &quot;15 + 20 equals 35.&quot;  (is true) Negation:  &quot;15 + 20 does not equal 35.&quot;  (is false) </li></ul><ul><li>  2. &quot;A dog is a cat.&quot;  is a false statement. &quot;A dog is not a cat.&quot; is a true statement. </li></ul>
9. 9. Compound Statement <ul><li>A compound statement can be formed from 2 given statements using the word ‘and’ or ‘or’. </li></ul><ul><li>Examples : </li></ul><ul><li>3 is a prime number </li></ul><ul><li>3 is an odd number </li></ul><ul><li>3 is a prime and odd number. </li></ul>
10. 10. Truth Value of a compound statement using ‘and’ <ul><li>a conjunction is a compound sentence formed by combining two statements (or facts) using the word &quot;and.&quot;   A conjunction is true only when BOTH statements (or facts) are true. </li></ul>
11. 11. <ul><li>Examples: </li></ul><ul><li>1.  &quot;Blue is a color and 7 + 3 = 10.&quot;  (T and T = T)      Since both facts are true, the entire sentence is true.   </li></ul><ul><li>2.  &quot;One hour = exactly 55 minutes and one minute = exactly 60 seconds.&quot;       (F and T = F)  Since the first fact is false, the entire sentence is false. </li></ul><ul><li>3.  &quot;3 + 4 = 6 and all dogs meow.&quot;  (F and F = F)      Since both facts are false, the entire sentence is false. </li></ul>
12. 12. Truth Value of a compound statement using ‘or’ <ul><li>a disjunction is a compound sentence formed by combining two statements (or facts) using the word &quot;or.&quot;  </li></ul><ul><li>A disjunction is true when EITHER or BOTH sentences (or facts) are true. </li></ul>
13. 13. <ul><li>Examples: </li></ul><ul><li>1.  &quot;Blue is a color or 7 + 3 = 10.&quot; (T or T = T)  Since both facts are true, the entire sentence is true.   </li></ul><ul><li>  2.  &quot;One hour = exactly 55 minutes or one minute = exactly 60 seconds.&quot; (F or T = T)  Since the second fact is true, the entire sentence is true. </li></ul><ul><li>  3.  &quot;3 + 4 = 6 or all dogs meow.&quot; (F or F = F) Since both facts are false, the entire sentence is false.   </li></ul><ul><li>4.  &quot;The word cat has 3 letters or the word dog has four letters.&quot;  (T  or F = T)  Since the first fact is true, the entire sentence is true. </li></ul>
14. 14. Online Activity <ul><li>http://www.regentsprep.org/Regents/math/tables/PracAnd.htm </li></ul>