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- https://math.libretexts.org/Courses/Los_Angeles_City_College/Math_230-Mathematics_for_Liberal_Arts_Students/05%3A_Logic/5.06%3A__Equivalent_StatementsBecause the last column is true for every entry, the biconditional is valid and the statement p → q p → q is logically equivalent to the statement ~ p ∨ q ~ p ∨ q . Symbolically, p → q “Lassie does no...Because the last column is true for every entry, the biconditional is valid and the statement p → q p → q is logically equivalent to the statement ~ p ∨ q ~ p ∨ q . Symbolically, p → q “Lassie does not like to bark” is ~ q ~ q and “Some dogs do not bark” is ~ p ~ p . The statement, “If Lassie does not like to bark, then some dogs do not bark,” has the form “if ~ q ~ q , then ~ p ~ p ,” which is the form of the contrapositive.
- https://math.libretexts.org/Courses/Los_Angeles_City_College/Math_230-Mathematics_for_Liberal_Arts_Students/08%3A_More_Probability/8.06%3A_Applications_of_the_Normal_DistributionFor the following exercises, recall that if we flip a coin at least 20 times, the distribution of the number of heads is approximately normal with mean equal to half the number of flips and standard d...For the following exercises, recall that if we flip a coin at least 20 times, the distribution of the number of heads is approximately normal with mean equal to half the number of flips and standard deviation equal to half of the square root of the number of flips.
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Contemporary_Mathematics_(OpenStax)/01%3A__Sets/1.02%3A_SubsetsSequences and series are defined as infinite subsets of the set of natural numbers by forming a relationship between the sequence or series in terms of a natural number, n n . For example, the set of ...Sequences and series are defined as infinite subsets of the set of natural numbers by forming a relationship between the sequence or series in terms of a natural number, n n . For example, the set of even numbers can be defined using set builder notation as { a | a = 2 n where n is a natural number } { a | a = 2 n where n is a natural number } . The formula in this case replaces every natural number with two times the number, resulting in the set of even numbers, { 2 , 4 , 6 , … } { 2 , 4 , 6 ,…
- https://math.libretexts.org/Courses/Northwest_Florida_State_College/NWFSC_MGF_1130_Text/04%3A_Set_Theory/4.02%3A_Basic_Set_ConceptsThe verbal description of the set is, “Set B B is the set of all elements b b such that b b is a ball.” This set can be written in set builder notation as follows: B = { b | b is a ball . } B = { b | ...The verbal description of the set is, “Set B B is the set of all elements b b such that b b is a ball.” This set can be written in set builder notation as follows: B = { b | b is a ball . } B = { b | b is a ball . } Because the set of integers is a subset of the set of rational numbers, and the set of integers is infinite, the set of rational numbers is also infinite.
- https://math.libretexts.org/Courses/Northwest_Florida_State_College/NWFSC_MGF_1130_Text/04%3A_Set_Theory/4.01%3A_IntroductionThe members of the group are the individual items in the drawer, such as a fork or a spoon. In statistical studies, a set is a well-defined collection of objects used to identify an entire population ...The members of the group are the individual items in the drawer, such as a fork or a spoon. In statistical studies, a set is a well-defined collection of objects used to identify an entire population of interest. For example, in a research study examining the effects of a new medication, there can be two sets of people: one set that is given the medication and a different set that is given a placebo (control group).
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Contemporary_Mathematics_(OpenStax)/07%3A_Probability/7.06%3A_Probability_with_Permutations_and_CombinationsThe number of possible starting hands is 100 C 7 = 16 , 007 , 560 , 800 100 C 7 = 16 , 007 , 560 , 800 . There are 100 − 44 = 56 100 − 44 = 56 consonants in the bag, so the number of all-consonant han...The number of possible starting hands is 100 C 7 = 16 , 007 , 560 , 800 100 C 7 = 16 , 007 , 560 , 800 . There are 100 − 44 = 56 100 − 44 = 56 consonants in the bag, so the number of all-consonant hands is 56 C 7 = 231 , 917 , 400 56 C 7 = 231 , 917 , 400 . Thus, the probability of drawing all consonants is 231 , 917 , 40 16 , 007 , 560 , 800 = 32 , 139 2 , 425 , 388 ≈ 0.0145 231 , 917 , 40 16 , 007 , 560 , 800 = 32 , 139 2 , 425 , 388 ≈ 0.0145 .
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Contemporary_Mathematics_(OpenStax)/06%3A_Money_Management/6.03%3A__Simple_InterestThe future value, FV FV , of an investment that yields simple interest is F V = P + I = P + P × r × t F V = P + I = P + P × r × t , where P P is the principal (amount invested at the start), r r is th...The future value, FV FV , of an investment that yields simple interest is F V = P + I = P + P × r × t F V = P + I = P + P × r × t , where P P is the principal (amount invested at the start), r r is the annual interest rate in decimal form, and t t is the length of time the money is invested.
- https://math.libretexts.org/Courses/Los_Angeles_City_College/Math_230-Mathematics_for_Liberal_Arts_Students/05%3A_Logic/5.03%3A__Compound_StatementsThe hypothesis of the biconditional statement is ~ ( r ∨ s ) ~ ( r ∨ s ) and is written in words as: “It is not the case that my child played video games or streamed a movie.” The conclusion of the bi...The hypothesis of the biconditional statement is ~ ( r ∨ s ) ~ ( r ∨ s ) and is written in words as: “It is not the case that my child played video games or streamed a movie.” The conclusion of the biconditional statement is ~ ( p ∧ q ) ~ ( p ∧ q ) , which translates to: “It is not the case that my child finished their homework and cleaned their room.” Because the biconditional, ↔ ↔ translates to if and only if, one possible translation of the statement is: “It is not the case that my child pla…
- https://math.libretexts.org/Courses/Los_Angeles_City_College/Math_230-Mathematics_for_Liberal_Arts_Students/05%3A_Logic/5.01%3A_IntroductionThe main goal of arguments made by lawyers is to convince a judge and jury that their arguments are valid and well-supported by the facts of the case, so the case should be ruled in their favor. Your ...The main goal of arguments made by lawyers is to convince a judge and jury that their arguments are valid and well-supported by the facts of the case, so the case should be ruled in their favor. Your ability to form and comprehend logical arguments is a valuable tool in many areas of life, whether you're planning a dinner date, negotiating the purchase of a new car, or persuading your boss that you deserve a raise.
- https://math.libretexts.org/Courses/Coalinga_College/Math_for_Educators_(MATH_010A_and_010B_CID120)/07%3A_Logic/7.09%3A_Chapter_Summary/7.9.05%3A_Chapter_ReviewIf the , \(p\), of a conditional statement is true, then the conclusion, \(q\), must also be true for the conditional statement \(p \rightarrow q\) to be true. Write the negation of the conditional st...If the , \(p\), of a conditional statement is true, then the conclusion, \(q\), must also be true for the conditional statement \(p \rightarrow q\) to be true. Write the negation of the conditional statement in words: If Thomas Edison invented the phonograph, then albums are made of vinyl, or the transistor radio was the first portable music device.
- https://math.libretexts.org/Courses/Coalinga_College/Math_for_Educators_(MATH_010A_and_010B_CID120)/07%3A_Logic/7.07%3A__De_Morgans_LawsBy De Morgan's Law, the negation of a conjunction, \(\sim(p \wedge q)\), is logically equivalent to \(\sim p \vee \sim q . \sim p\) is "Kristen is not a biomedical engineer," and \(\sim q\) is "Thomas...By De Morgan's Law, the negation of a conjunction, \(\sim(p \wedge q)\), is logically equivalent to \(\sim p \vee \sim q . \sim p\) is "Kristen is not a biomedical engineer," and \(\sim q\) is "Thomas is not a chemical engineer." The columns will be \(p, q, \sim p, \sim q, p \wedge q, \sim(p \wedge q), \sim p \vee \sim q\), and the biconditional statement is \(\sim(p \wedge q) \leftrightarrow \sim p \vee \sim q\).
