2.3 Propositions, Negations, Truth Tables

Propositions are statements that may be true or false.

Comments, opinions and questions are not propositions

Letters are used to represent propositions in mathematical logic.

If p is used to represent a proposition, is used to represent the negation of that proposition.

Let p represent the proposition ‘it is raining’. would represent ‘it is not raining’

Ex 1 Find the negation of

(a) x is a dog (b) ,

Compound Propositions: statements which are formed using connectives such as and and or.

Symbolic Logic

Ex 2

(a) For breakfast I will have cereal and toast.

is the symbol for and/conjunction.

The proposition can be written as where p: I will have cereal q: I will have toast

(b) For lunch I will have cereal or toast.

is the symbol for or/disjunction.

The proposition can be written as where p: I will have cereal q: I will have toast

Conjunctions are true when both of its propositions are true.

Disjunctions can have two forms: inclusive and exclusive. Inclusive disjunctions are true if one or both of their propositions are true. Exclusive disjunctions are true when only one of their propositions are true. The symbol for exclusive disjunction is v

By default, disjunctions are considered inclusive unless the language tells you otherwise.

**Links to Venn Diagrams**

Truth Tables

When creating a truth table you need rows, where n is the number of propositions in your statement. If you have 2 propositions, you need 4 rows.

Ex 3 Create a set of truth tables for conjunction, inclusive disjunction and exclusive disjunction.

Logical Equivalence

If two compound propositions have the same results on a truth table, we say that the two propositions are logically equivalent.

Ex 4 Show that using a truth table and a Venn Diagram.

Tautologies are compound statements that are always true.

Contradictions are compound statements that are always false.

p. 496# 1-7

p. 499# 1-3

p. 501# 1-10

p. 504# 1-8