Ch2_Logic - MA100, Fall 2010, PART 1: Precalculus [2]...

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MA100, Fall 2010, PART 1: Precalculus [2] Elements of Logic 1. Statements in general Definition In math, a statement is a declarative sentence that can be true or false, but not both. Example: “4 is an even number” is a true statement. Example: “The Earth year has 370 days” is a false statement. Example: Indicate the truth value for the following statement: “6 is a prime number”. Solution: The statement “6 is a prime number” is false since 3 is a divisor of 6. (Recall that a natural number is prime if its only positive divisors are 1 and itself.) Example: Indicate the truth value for the following statement: For any z Z , we have 15 2 z + 1 Z . Solution: The given statement is false, since for z = 3 we have that 15 2 z + 1 = 15 2 × 3 + 1 = 15 10 = 1 . 5 / Z . Definition A mathematical identity is an equality relation that is always true (for all values of the variables in the relation). Sometimes we are asked to verify that certain identities are indeed true. Example: Verify that ( x + 3) 2 = x 2 + 6 x + 9. Solution: We have to verify that the left hand side (denoted LHS), ( x +3) 2 , of the above relation always equals the right hand side (denoted RHS), x 2 + 6 x + 9. One way to do this is to start with the LHS and calculate: LHS = ( x + 3) 2 = ( x + 3)( x + 3) = x 2 + 3 x + 3 x + 9 = x 2 + 6 x + 9 = RHS . Another way to write the verification of the identity ( x +3) 2 = x 2 +6 x +9 is by using the symbol for logical equivalence, ⇐⇒ , as follows: ( x + 3) 2 = x 2 + 6 x + 9 ⇐⇒ ( x + 3)( x + 3) = x 2 + 6 x + 9 ⇐⇒ x 2 + 3 x + 3 x + 9 = x 2 + 6 x + 9 ⇐⇒ x 2 + 6 x + 9 = x 2 + 6 x + 9 ⇐⇒ 0 = 0 which is true. 1
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Definition An identity that is used often is called a formula . Definition An equation is an equality relation depending on some variables (also called “un- knowns”) for which we are asked to find all values that make the two sides of the equation equal. In plain language, an equation is a math question for which we must find one or more correct answers. For instance, when we solve the equation x - 2 = 0 for x , we are answering to the question “What should be x so that when substituted into x - 2 = 0 we get a true statement”? The correct answer is obviously x = 2 , because if we substitute 2 for x in x - 2 = 0, we get the true statement 2 - 2 = 0, and no other value of x will give a true statement. (For example, if we substitute 3 for x we get the false statement 3 - 2 = 0.) Note An equation is not a mathematical identity! An equation is essentially a question, whereas an identity is a statement that is always true. Example: Solve 4( x - 2) + 5( x + 2) = x - 6 . Solution: We have to find all possible values for x that give a true statement. 4(
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Ch2_Logic - MA100, Fall 2010, PART 1: Precalculus [2]...

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