CK-12 Geometry - Second Edition

http://www.ck12.org Chapter 2. Reasoning and Proof

Truth tables use these symbols and are another way to analyze logic.

First, let’s relatepand∼p. To make it easier, setpas:An even number.

Therefore,∼pisAn odd number. Make a truth table to find out if they are both true. Begin with all the “truths” of
p, true (T) or false (F).

TABLE2.5:

p

T

F

Next we write the corresponding truth values for∼p.∼phas the opposite truth values ofp. So, ifpis true, then
∼pis false and vise versa.

TABLE2.6:

p ∼p

T F

F T

Example 11:Draw a truth table forp,qandp∧q.

Solution:First, make columns forpandq. Fill the columns with all the possible true and false combinations for the
two.

TABLE2.7:

p q

T T

T F

F T

F F

Notice all the combinations ofpandq.Anytime we have truth tables with two variables, this isalways how we
fill out the first two columns.

Next, we need to figure out whenp∧qis true, based upon the first two columns.p∧qcan only be true if BOTHp
andqare true.So, the completed table looks like this:

This is how a truth table with two variables and their “and” column is always filled out.

Example 12:Draw a truth table forp,qandp∨q.

Solution:First, make columns forpandq, just like Example 11.