Different compound propositions

Since propositions in english can be ambiguous in how its meaning truth tables, provide an unambiguous way to show the meaning of a compound proposition.

Truth Tables

A circuit through which the wires carry T/F values and the gates hold logical operations (and : $$\land$$  ;  or : $$\lor$$  ;  not : $$\lnot$$  ;  etc.) .

You should ask:
- What is the circuit doing?
- Can it be simplified?

Logic Circuits

Built from multiple propositions using logical operators to connect the propositions

University of Michigan - Ann Arbor

A declarative statement that is either true or false

Propositions (mathematics)

University of Michigan EECS 203 Lecture Notes

- 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

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.

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