math 10

• Algebraic expressions can be simplified by combining like terms and applying the order of operations.
• Logic plays a very important role in mathematics, serving as the foundation on which the discipline is built.
• Mathematics can be considered as a language with its own symbols and "grammar", where these symbols may represent various mathematical objects like numbers, sets, or functions.
• The grammar of mathematics is the rules when combining these symbols.
• Logic is used to deduce properties of these objects and rules based on some axioms.
• Logic is just as important in our everyday life, enabling us to distinguish between true and false statements and arguments.
• In an era of fake news, post-truths, and false advertising, it is crucial to be able to discern what is true or false.
• Learning Outcomes after studying this module include determining whether a statement has truth value, negating simple and compound statements, describing the various forms of the conditional, using truth tables to determine the truth value of a statement, determining whether an argument is valid or invalid using Euler diagrams or truth tables, and illustrating deductive and inductive reasoning.
• We can use Euler diagrams to determine the validity of an argument.
• An argument is invalid if being hairy does not automatically imply being a dog.
• If an argument is valid, there should only be one possible conclusion.
• The argument p → q is valid if the premises are assumed to be true, then the conclusion must also hold true.
• The premises are assumed to be true although the statements may not be true in the strict sense.
• One may also show that an argument is invalid by exhibiting two different diagrams representing the premises.
• To show an argument is invalid, it suffices to exhibit an Euler diagram satisfying all the premises but not the given conclusion.
• An argument is valid if the conclusion is satisfied by the Euler diagram representing all premises.
• An argument consists of premises, say p1, p2, ..., pn, and a conclusion q.
• The fallacy of the inverse is represented by the conditional { [( p → q ) ∧ ~ p ] → ~ q }, where p and q are propositions and the argument is false when p is false and q is true.
• Using Euler Diagrams, establish the validity of the modus pones, modus tollens and syllogism.
• All scientists contribute to our country’s economic growth.
• The fallacy of ad populum occurs when an argument is assumed to be valid since many people believe in it.
• The fallacy of the converse is represented by the conditional { [( p → q ) ∧ q ] → p }, where p and q are propositions and the argument is false when p is true and q is false.
• Fallacies include the fallacy of the converse and the fallacy of the inverse.
• The fallacy of ad hominem occurs when an argument is based on the character of the opponent instead of the argument itself, and may involve insulting the opponent to make the opponent's argument seem false.
• Fallacies can also be of other forms, such as the fallacy of ad hominem and the fallacy of ad populum.
• A mathematical statement is a statement that can be assigned a truth value and classified as true or false, but not both.
• Lowercase letters, p, q, r, s,...., are used to represent mathematical statements.
• Examples of mathematical statements include p: 1 + 1 = 2, q: 2 + 3 = 6, r: All roses are red, s: The Philippines has more than 7,100 islands.
• Mathematical statements are declarative sentences that are either true or false, but not both.
• Declarative statements whose truth value is not clear or a matter of opinion are not considered as mathematical statements.
• Questions, exclaimations, and imperatives are not considered as mathematical statements as well, since these sentences do not have a truth value.
• The boat to be used can only accommodate Juan and either the cat, mouse or the sack of rice.
• If left together, the cat will eat the mouse.
• Jose must take a cat, a mouse, and a sack of rice across a river with his boat.
• The child in front (the shortest) is holding neither a red nor blue balloon.
• Susie is holding a red balloon.
• The child holding the blue balloon is right in front of the child with the yellow balloon.
• Amy is in front of Tessie.
• Members of Tribe T always tell the truth, Tribe L members never tell the truth and Tribe X members sometimes tell the truth and they sometimes lie.
• One child is holding a white balloon.