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D420 Discrete Math: Logic

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Discrete Math: Logic D420

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78 terms Zac Exclusive or. ⊕One or the other, but not both.We can go to the park or the movies.

inclusive or is a:disjunction

Order of operations in absence of parentheses. 1. ¬ (not)

• ∧ (and)
• ∨ (or)

the rule is that negation is applied first, then conjunction, then disjunction:

truth table with three variablessee pic 2^3 rows propositionp → q

Ex: If it is raining today, the game will be cancelled.

Converse:q → p

If the game is cancelled, it is raining today.Contrapositive¬q → ¬p If the game is not cancelled, then it is not raining today.

Inverse:¬p → ¬q

If it is not raining today, the game will not be cancelled.

biconditionalp ↔ q true when P and Q have the same truth value.see truth table pic.free variableex.P(x) the variable is free to take any value in the domain bound variable∀x P(x) bound to a quantifier.In the statement (∀x P(x)) ∧ Q(x),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. ¬∀x P(x) ≡ ∃x ¬P(x) ¬∃x P(x) ≡ ∀x ¬P(x) using a truth table to establish the validity of an argument 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.see pic.theoremany statement that you can prove proofA 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: which are statements assumed to be true.proofs by exhaustiontrying everything in the given universe.

proofs by counter exampleshow 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 proofsused for conditional statements If p then q Assume p Therefore q proofs by contrapositiveproves 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(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 casesA 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 object in a set are calledelements

The symbol ∈ is used:to indicate that an element is in a set, as in 2 ∈ A

The set with no elements is called the empty set and is denoted by the symbol ∅.The empty set is sometimes referred to as the null setand can also be denoted by {}. Because the empty set has no elements, for any element a, a ∉ ∅ is true.

the cardinality of the empty set is:zero

NThe set of natural numbers: All integers greater than or equal to 0. 0, 1, 2, ...ZThe set of all integers. ..., -2, -1, 0, 1, 2, ...QThe set of rational numbers: All real numbers that can be expressed as a/b, where a and b are integers and b ≠ 0. 0, 1/2, 5.23, -5/3 RThe set of real numbers.

The superscript + is used to indicate the positive elements of a particular set.For example, the set R+ is the set of all positive real numbers, and Z+ is the set of all positive integers.The universal set, usually denoted by the variable U, is a set that contains all elements mentioned in a particular context.The empty set ∅ is not the same as { ∅ }.The cardinality of { ∅ } is one since it contains exactly one element, which is the empty set.If every element in A is also an element of B, then A is a subset of B, denoted as A ⊆ B If there is an element of A that is not an element of B, then A is not a subset of B, denoted as A ⊈ B.Two sets are equal if and only if each is a subset of the

other:

A = B if and only if A ⊆ B and B ⊆ A If A ⊆ B and there is an element of B that is not an

element of A (i.e., A ≠ B):

then A is a proper subset of B, denoted as A ⊂ B.Proper subset is a strictly smaller subset, must be smaller.Sets of sets.A = { { 1, 2 }, ∅, { 1, 2, 3 }, { 1 } } The set A has four elements: { 1, 2 }, ∅, { 1, 2, 3 }, and { 1 }. For example, { 1, 2 } ∈ A.Note that 1 is not an element of A, so 1 ∉ A, although { 1 } ∈ A. Furthermore, { 1 } ⊈ A since 1 ∉ A.power setsThe power set of a set A, denoted P(A), is the set of all subsets of A. For example,

if A = { 1, 2, 3 }, then:

P(A) = { ∅, { 1 }, { 2 }, { 3 }, { 1, 2 }, { 1, 3 }, { 2, 3 }, { 1, 2, 3 } }

Since |A| = 3, the cardinality of the power set of A is: 2^3 = 8

set operations intersectionA intersects B or A ∩ B And.Union.A U B. both.Or.sets with multiple operationsFor example, the set A ∪ ( B ∩ C ) is the union of the set A and the set B ∩ C.Set difference and symmetric difference.Difference of set A and set B is A - B or the set of elelemnts in A but not in B.Symmetric difference is communitive.Denoted A ⊕ B. the set of elements A and B but not both.Complement of A is the set of all elements in the U not elements of A. sometimes U - A see pic.set identities and lawssee pic.