- Two parallel lines on a plane have a point in common : (FALSE)

- Ann Arbor is the capital of America : (FALSE)

- 2 + 3 + 5 = 10 : (TRUE)

Examples of propositions

Propositions and connectives are used to make compound propositions. One can make deductions and proofs drawing from the structure of propositions

Propositional Logic

Built from multiple propositions using logical operators to connect the propositions

Compound proposition

If the proposition is not a contradiction then it is satisfiable, otherwise it is not.

How can we know if a proposition is satisfiable?

If two compound propositions have the same truth table they are logically equivalent (p $$\equiv$$ q).
- ((p $$\rightarrow$$ q) $$\land$$ (q $$\rightarrow$$ p) $$\equiv$$ p $$\leftrightarrow$$ q is a tautology)

Logically Equivalent

A proposition that is always false (e.g. $$\lnot$$ p $$\land$$ p)

Contradiction (propositions)

A proposition that is always true (e.g. $$\lnot$$ p $$\lor$$ p)

Tautology (propositions)

Axioms (propositions)

A chain of logical deductions that leads to a proposition form an original (base) set of axioms.

Formal Proofs (of propositions)

A predicate is an open declarative statement containing one or more variables. It becomes a proposition when every variable is assigned an object or value from its domain or is bound by a quantifier, so that the resulting statement can be true or false.

Predicate (mathematical logic)

A literal is a propositional variable, such as p, or its negation, NOT p. A truth assignment gives p either the value true or false; NOT p has the opposite value.

Literal in propositional logic

A declarative statement that is either true or false

University of Michigan - Ann Arbor

Basic concepts in Discrete math

University of Michigan EECS 203 Lecture Notes

an inferential argument for a mathematical statement that uses assumptions to logically show the conclusion is valid

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