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## Homework

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**Homework**Review notes Complete Worksheet #2**Homework**State whether the conditional sentence is true or false 1. If 1 = 0, then 1 = – 1 True F F T**Homework**Give the converse of the conditional sentence and state if it is sometimes, always, or never true. 5. If two triangles are congruent, then their corresponding angles are congruent. If the corresponding angles of two triangles are congruent, then the triangles are congruent. Sometimes true.**Homework**Give the converse of the conditional sentence and state if it is sometimes, always, or never true. 9. If ab < ac, then b < c If b < c, then ab < ac, sometimes true**Homework**State the contrapositive for each conditional sentence. 13. If an angle of a parallelogram is a right angle, then the parallelogram is a rectangle. If a parallelogram is not a rectangle, then no one of its angles is a right angle.**Homework**State the assumptions that would have to be made if the given statement is to be proven by contradiction. 17.**Homework**State each conditional sentence in the if-then form. 21.**Homework**State each conditional sentence in the if-then form. 25.**Homework**Name the axiom, theorem, or definition that justifies each step. 1. Proof: Definition of Subtraction Associative Axiom Additive Inverses Additive Identity Transitive Property**Homework**Solve over 5.**Homework**Solve over 9.**Homework**State whether each set is closed under (a) addition and (b) multiplication. If not, give an example. 13. N (a) Closed (b) Closed**Foundations of Real Analysis**Conditional Sentences Addition and Multiplication Properties of Real Numbers**Order**The real numbers are ordered by the relation less than (<). a < b if the graph of a is to the left of the graph of b on the number line. Less than (<) is an undefined relation.**Axiom of Comparison**For all real numbers a and b,**Transitive Axiom of Order**For all real numbers a, b and c, if a < b and b < c, then a < c**Addition Axiom of Order**For all real numbers a, b and c, if a < b, then a + c < b + c**Positive and Negative**A real number, x, is positive if x > 0 and negative if x < 0**Multiplication Axiom of Order**For all real numbers a, b and c, if 0 < c and a < b, then ac < bc.**Sign Definitions**• a > b if and only if b < a • a ≥ b if and only if a > b or a = b • a ≤ b if and only if a < b or a = b**Theorem Three**For all real numbers a, b, and c:**Theorem Four**For all real numbers a, b, and c, if c < 0 and a < b, then ac > bc**Theorem Five**For all real number a and b: • ab > 0 if and only if a and b are both positive or a and b are both negative. • ab < 0 if and only if a and b are opposite in sign.**Example**Solve the inequality and graph each non-empty solution. 2. -----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----**Example**Solve the inequality and graph each non-empty solution. 6. -----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----**Example**Solve the inequality and graph each non-empty solution. 10. -----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----**Example**Solve the inequality and graph each non-empty solution. 14. -----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----**Example**18. Solve and graph the intersection of the sets: -----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----**Absolute Value**Definition – For all real numbers x: This may also be written as a piecewise function:**Theorem Six**If a ≥ 0, |x| = a if and only if x = –a or x = a. If a > 0, |x| < a if and only if –a < x < a If a > 0, |x| > a if and only if x < –a or x > a**Theorem Seven**For all real numbers a, |a|2 = a2**Theorem Eight**For all real numbers a and b,**Example**Find and graph each non-empty set over R. 2. -----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----**Example**Find and graph each non-empty set over R. 6. -----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----**Example**Find and graph each non-empty set over R. 10. -----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----**Example**Find and graph each non-empty set over R. 18. -----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----**Example**Find and graph each non-empty set over R. 22. -----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----**Example**Solve. 26.**Natural Numbers**The set of all positive integers {1, 2, 3,…}**Factor or Divisor**a is a factor of b (symbolically, a|b) if there is an integer c such that ac = b**Prime Number**Any number greater than 1 that has only one and itself as factors. Sometimes called a prime.**Composite**Any integer greater than 1 that is not prime**Theorem Nine**The Fundamental Theorem of Arithmetic – Every integer greater than 1 can be expressed as a product p1p2p3…pn in which p1, p2, p3, …, pn are primes. Furthermore, the factorizing is unique, except for the order in which primes are written.**Greatest Common Factor**The greatest common factor or GCF of a and b is the largest positive integer that is a factor of both a and b**Relatively Prime**Integers whose GCF is 1**Theorem Ten**The Division Algorithm – Given integers s and t, t > 0, there exist unique integers q and r such that s = tq + r and 0 ≤ r < t**Euclidean Algorithm**To find the GCF, divide the larger by the smaller, then the divisor by the remainder and so on until the remainder is zero; the last divisor is the GCF For example:**Irrational Numbers**All real numbers that are not rational, e.g., π and e, the base for natural logarithms**Theorem Eleven**There is no rational number whose square is 2.