discrete math 1 wgu

Exclusive or. ⊕ - answer-One or the other, but not both. We can go to the park or the movies. inclusive or is a: - answer-disjunction Order of operations in absence of parentheses. - answer-1. ¬ (not) 2. ∧ (and) 3. ∨ (or) the rule is that negation is applied first, then conjunction, then disjunction: truth table with three variables - answer-see pic 2^3 rows proposition - answer-p → q Ex: If it is raining today, the game will be cancelled. Converse: - answer-q → p If the game is cancelled, it is raining today. Contrapositive - answer-¬q → ¬p If the game is not cancelled, then it is not raining today. Inverse: - answer-¬p → ¬q If it is not raining today, the game will not be cancelled. biconditional - answer-p ↔ q true when P and Q have the same truth value. see truth table pic. free variable - answer-ex. P(x) the variable is free to take any value in the domain bound variable - answer-∀x P(x) bound to a quantifier. In the statement (∀x P(x)) ∧ Q(x), - answer-the variable x in P(x) is bound the variable x in Q(x) is free. this statement is not a proposition cause of the free variable. summary of De Morgan's laws for quantified statements. - answer-¬∀x P(x) ≡ ∃x ¬P(x) ¬∃x P(x) ≡ ∀x ¬P(x) using a truth table to establish the validity of an argument - answer-see pic. In order to use a truth table to establish the validity of an argument, a truth table is constructed for all the hypotheses and the conclusion. A valid argument is a guarantee that the conclusion is true whenever all of the hypotheses are true. If when the hypotheses are true, the conclusion is not, then it is invalid. the argument works if every time the hypotheses (anything above the line) are true, the conclusion is also true. hypotheses dont always all need to be true, see example. but every time all the hypotheses are true, the conclusion needs to be true as well. rules of inference. - answer-see pic. theorem - answer-any statement that you can prove proof - answer-A proof consists of a series of steps, each of which follows logically from assumptions, or from previously proven statements, whose final step should result in the statement of the theorem being proven. the proof of a theorem may make use of axioms: - answer-which are statements assumed to be true. proofs by exhaustion - answer-trying everything in the given universe. proofs by counter example - answer-show that one fails. A counterexample is an assignment of values to variables that shows that a universal statement is false. A counterexample for a conditional statement must satisfy all the hypotheses and contradict the conclusion. direct proofs - answer-used for conditional statements If p then q Assume p Therefore q proofs by contrapositive - answer-proves a conditional theorem of the form p → q by showing that the contrapositive ¬q → ¬p is true. In other words, ¬c is assumed to be true and ¬p is proven as a result of ¬q. Logically equivalent to if p then q proof by contradiction - answer-(indirect proof) starts by assuming that the theorem is false and then shows that some logical inconsistency arises as a result of this assumption. Notice not a conditional. Want to prove Y Assume not Y Find a contradiction in X and Not Y Therefore, claim not not Y. proof by cases - answer-A proof by cases of a universal statement such as ∀x P(x) breaks the domain for the variable x into different classes and gives a different proof for each class. Every value in the domain must be included in at least one class. Unit 2 sets and functions - answer- object in a set are called - answer-elements The symbol ∈ is used: - answer-to indicate that an element is in a set, as in 2 ∈ A