Discrete Math

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∼p

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not p

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p ⊕ q or p XOR q

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p or q but not both p and q

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P ≡ Q

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P is logically equivalent to Q

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p → q

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if p then q

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p ↔ q

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p if and only if q

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∴

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therefore

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P(x) ⇒ Q(x)

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every element in the truth set for P(x) is in the truth set for Q(x)

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P(x) ⇔ Q(x)

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P(x) and Q(x) have identical truth sets

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∀

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for all

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∃

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there exists

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AND-gate

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NAND-gate

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NOR-gate

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NOT-gate

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OR-gate

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|

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Sheffer stroke (NAND)

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↓

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Peirce arrow (NOR)

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a ∈ A

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a is an element of A

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Set-Builder Notation

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a is not an element of A

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{a1, a2,..., an }

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the set with elements a1, a2,..., an

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{x ∈ D | P(x)}

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the set of all x in D for which P(x) is true

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the sets of all real numbers, negative real numbers, positive real numbers, andnonnegative real numbers

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the sets of all integers, negative integers, positive integers, and nonnegative integers

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the sets of all rational numbers, negative rational numbers, positive rational numbers,and nonnegative rational numbers

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N

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the set of natural numbers

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A is a subset of B

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A is not a subset of B

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A union B

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A intersect B

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the Cartesian product of A1, A2,..., An

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∅

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the empty set

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universal statement

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a certain property is true for all elements in a set.

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conditional statement

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if one thing is true then some other thing also hasto be true.

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universal existential statement

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a statement that is universal because its first part says that a certain property is true for all objects of a given type, and it is existential because its second part asserts the existence of something

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existential universal statement

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a statement that is existential because its first part asserts that a certain object exists and is universal because its second part says that the object satisfies a certain property for all things of a certain kind

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statement (or proposition)

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a sentence that is true or false but not both

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conjunction of p and q

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p ∧ q

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disjunction of p and q

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p ∨ q

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negation of p

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∼p

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statement form (or propositional form)

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an expression made up of statement variables (such as p, q, and r) and logical connectives (such as ∼, ∧, and ∨) that becomes a statement when actual statements are substituted for the component statement variables

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tautology

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a statement form that is always true regardless of the truth values ofthe individual statements substituted for its statement variables

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contradication

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a statement form that is always false regardless of the truth values of the individual statements substituted for its statement variables

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contrapositive

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The contrapositive of p → q is ∼q → ∼p.

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syllogism

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An argument form consisting of two premises and a conclusion

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modus ponens

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If p then q.

p

∴ q

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modus tollens

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If p then q.

∼q

∴ ∼p

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fallacy

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an error in reasoning that results in an invalid argument

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Boolean expression

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An expression composed of Boolean variables and the connectives ∼, ∧, and ∨