Using truth tables on large propositions would be very time consuming and difficult. This is why it is better to use logical identities which you can use to prove equivalences. They include:
- Distributive laws
- De Morgan's laws
- Associative laws
- Commutative  laws
- Double-negation law
- Identity laws
- Domination laws
- Idempotent Laws
- Implication laws
- Contrapositive law

Logical Identities (proving logical equivalences)

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)

University of Michigan - Ann Arbor

A declarative statement that is either true or false

Propositions (mathematics)

University of Michigan EECS 203 Lecture Notes

http://people.math.gatech.edu/~ecroot/2406_2012/basic_logic.pdf

Propositional Equivalences Worksheet

- 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?

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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