Logical Reasoning

1. Name ________________________
Worksheet 1.2
Logic and Truth Tables
We can represent the truth of expressions in a tabular form called “truth tables.” These tables
consider all cases and can add great insight into otherwise complicated expressions. Here are
examples of some of most basic truth tables.
The truth table for negation (“not”)
p ¬ p
T F
F T
The truth table for disjunction (“or”)
p q p  q
T T T
T F T
F T T
F F F
1. Complete the following truth tables
(a) Truth table for conjunction (“and”)
p q p  q
T T
T F
F T
F F
(b) Truth table for biconditional (“if and only if”)
p q p ↔ q
T T
T F
F T
F F
Why do we say that the biconditional defines “logical equivalence”?

2. 2. The truth table for conditional (“if...then...”) can be quite confusing. Here it is.
p q p → q
T T T
T F F
F T T
F F T
Consider the example “For every integer n, if n > 2 then n2 > 4.” In this case p represents
“n > 2” and q represents “n2 > 4.” Thus if n = 3 then p is true and q is true.
(a) Find another value of n that makes p true and q true.
(b) Find a value for n that makes p false and q false.
(c) Find a value for n that makes p false and q true.
(d) Find a value for n that make p true and q false. Explain any problems you have. Why
are you having these problems?
The third row, corresponding to part (c), frustrates many people—you are not alone. The
truth has a special name. It is called “vacuously true.”
3. Complete the following truth tables
(a)
p q (¬ p) ↔ q
T T
T F
F T
F F
(b)
p q (¬ p)  q
T T
T F
F T
F F
(c)
p q r q → r p → (q → r)
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F

3. 4. Draw up and complete the truth tables for the following expressions. Please follow all of the
conventions that you have observed above (such as listing p before q, etc.)
(a) p  (¬ p)
(b) (¬ p)  (¬ q)
5. An expression is a tautology if it is always true, regardless the truth of its component
statements. Thus, the last column in its truth table will be all T’s. A statement is a contradiction
if the last column is always F—it is never true.
(a) Show that (p  q) → p is a tautology by completing this truth table.
p q p  q (p  q) → p
T T
T T
T F
T F
F T
F T
F F
F F
(b) Show that p  (¬ p) is a contradiction.
p p  (¬ p)
T
F

4. 6. Logically equivalent expressions have the same truth tables. Show that the following pairs of
statements are logically equivalent.
(a) (¬ p)  q and (p  q) (You saw this on the previous worksheet. Where?)
(b) p → q and (¬ q) → (¬ p) (The second statement is called the Contrapositive)
(c) (p  q)  r and (p  r)  (q  r) (Distribution of sorts)
(d) ¬ (p  q) and (¬ p)  (¬ q) (This, and part (e) is called De Morgan’s Law)
(e) ¬ (p  q) and (¬ p)  (¬ q)