LOGICAL EQUIVALENCES and
PRACTICE WITH TRUTH TABLES

A logical equivalence is a statement that two mathematical sentence forms are completely interchangeable:
if one is true, so is the other; if one is false, so is the other.

For example, we could express that an implication is equivalent to its contrapositive in either of the following ways:

[beautiful math coming... please be patient] $A \Rightarrow B\,$     is (logically) equivalent to     $\,(\text{not }B) \Rightarrow (\text{not }A)$

or

‘$(A \Rightarrow B) \ \ \Longleftrightarrow \ \ ((\text{not }B) \Rightarrow (\text{not } A))$’     is a logical equivalence

In this section, you will construct truth tables involving implications, equivalences, negations,
and the mathematical words ‘and’ and ‘or’.
In the process, you'll be introduced to many useful logical equivalences,
and will start to develop intuition for investigating logical equivalences.

NEGATING ‘and’ and ‘or’ SENTENCES:   DE MORGAN'S LAWS

How can a sentence ‘$A \text{ and } B\,$’ be false?
The only time an ‘and’ sentence is true is when both subsentences are true.
So, an ‘and’ sentence is false when at least one of the subsentences is false.

Precisely, the truth table below shows that:

[beautiful math coming... please be patient] $\text{not}(A \text{ and } B)$       is equivalent to       [beautiful math coming... please be patient] $ (\text{not } A) \text{ or } (\text{not } B)$

Here's the intuition you should have when looking at this sentence.
Start with (1) and work your way to (6):

(2) ... is false ... (1) an ‘and’ sentence ... (3) ... when ... (4) ... $\,A\,$ is false ... (5) ... or ... (6) ... $\,B\,$ is false
[beautiful math coming... please be patient] $\overbrace{\text{not}}$ $\overbrace{(A\text{ and }B)}$ $\overbrace{\text{is equivalent to}}$ $\overbrace{(\text{not }A)}$ $\overbrace{\text{ or }}$ $\overbrace{(\text{not }B)}$

In particular, note that you say ‘is false’ when you see the word ‘not’.

$A$ $B$ $A \text{ and } B$ [beautiful math coming... please be patient] $\text{not}(A \text{ and } B)$ $\text{not } A$ $\text{not } B$ $(\text{not } A) \text{ or } (\text{not } B)$ $(\text{not}(A \text{ and } B))\iff ((\text{not } A) \text{ or } (\text{not } B))$
T T T F F F F T
T F F T F T T T
F T F T T F T T
F F F T T T T T

Here's the critical observation:
the columns for ‘$\,\text{not}(A \text{ and } B)\,$’ and ‘$\,(\text{not } A) \text{ or } (\text{not } B)\,$’ are identical!
Some people like to think of this as a sort of distributive law, except that ‘and’ changes to ‘or’ in the distribution process.

For simplicity, in the subsequent truth tables we will not include the last column—the one that shows that the ‘is equivalent to’ statement is always true.
This last column takes up a lot of space, and makes things look more complicated than they really are.
It is equally convincing to compare the two relevant columns, to show that they are identical.

IMPORTANCE OF ASSOCIATIVE LAWS

Think about this:
the reason that we can write things like [beautiful math coming... please be patient] $\,1 + 2 + 3\,$ without ambiguity,
is because both $\,(1 + 2) + 3\,$ and $\,1 + (2 + 3)\,$ give the same result.
If they produced different results, then we'd have to write the parentheses every time we worked with these expressions.

You'll prove in the exercises that:

[beautiful math coming... please be patient] $(A \text{ or } B) \text{ or } C\,$       is equivalent to       $\,A \text{ or } (B \text{ or } C)$

Below, it is shown that:
[beautiful math coming... please be patient] $(A \text{ and } B) \text{ and } C\,$       is equivalent to       $\,A \text{ and } (B \text{ and } C)$

Thus (yeh!) we can write   ‘$\,A \text{ or } B \text{ or } C\,$’   and   ‘$\,A \text{ and } B \text{ and } C\,$’   without ambiguity.

Note that the truth table below involves three subsentences ($\,A\,$, $\,B\,$, and $\,C\ $),
so there are [beautiful math coming... please be patient] $\,2^3 = 8\,$ rows needed to cover all possible truth values.
Always list the eight rows in exactly the order that is shown here.

$A$ $B$ $C$ $A \text{ and } B$ [beautiful math coming... please be patient] $(A \text{ and } B) \text{ and } C$ $B \text{ and } C$ $A \text{ and } (B \text{ and } C)$
T T T T T T T
T T F T F F F
T F T F F F F
T F F F F F F
F T T F F T F
F T F F F F F
F F T F F F F
F F F F F F F

In the exercises, you will construct truth tables to prove all the following logical equivalences.

LOGICAL EQUIVALENCES
NAME/DESCRIPTION [beautiful math coming... please be patient] $1^{\text{st}}\,$ sentence ... is equivalent to ... $2^{\text{nd}}\,$ sentence INTUITION
de Morgan's Law:
negating an ‘and’ sentence
[beautiful math coming... please be patient] $\text{not}(A \text{ and } B)$ ... is equivalent to ... $(\text{not }A) \text{ or } (\text{not }B)$ an ‘and’ sentence is false
when
$\,A\,$ is false or $\,B\,$ is false
de Morgan's Law:
negating an ‘or’ sentence
[beautiful math coming... please be patient] $\text{not}(A \text{ or }B)$ ... is equivalent to ... $(\text{not }A) \text{ and }(\text{not }B)$ an ‘or’ sentence is false
when
$\,A\,$ is false and $\,B\,$ is false
associativity of ‘and’ [beautiful math coming... please be patient] $(A \text{ and }B) \text{ and }C$ ... is equivalent to ... $A \text{ and }(B \text{ and }C)$ the grouping doesn't matter
in an ‘and’ sentence
associativity of ‘or’ [beautiful math coming... please be patient] $(A \text{ or }B) \text{ or }C$ ... is equivalent to ... $A \text{ or }(B \text{ or }C)$ the grouping doesn't matter
in an ‘or’ sentence
commutativity of ‘and’ [beautiful math coming... please be patient] $A \text{ and } B$ ... is equivalent to ... $B \text{ and } A$ the order doesn't matter in an ‘and’ sentence
commutativity of ‘or’ $A \text{ or } B$ ... is equivalent to ... $B \text{ or } A$ the order doesn't matter in an ‘or’ sentence
law of double negation $\text{not}(\text{not }A)$ ... is equivalent to ... $A$ negating twice in succession
gets you back to where you started
distributive law [beautiful math coming... please be patient] $A \text{ or }(B \text{ and }C\,)$ ... is equivalent to ... $(A \text{ or }B) \text{ and }(A \text{ or }C\,)$ similar to: $\,a(b + c) = ab + ac$
distributive law [beautiful math coming... please be patient] $A \text{ and }(B \text{ or }C\,)$ ... is equivalent to ... $(A \text{ and }B) \text{ or }(A \text{ and }C\,)$ similar to: $\,a(b + c) = ab + ac$
alternate form
of an implication
$A \Rightarrow B$ ... is equivalent to ... $(\text{not }A) \text{ or }B$ an implication is true when
the hypothesis is false or
the conclusion is true
contrapositive
of an implication
[beautiful math coming... please be patient] $A \Rightarrow B$ ... is equivalent to ... $(\text{not }B) \Rightarrow (\text{not }A)$ an implication is equivalent to
its contrapositive
negating an implication [beautiful math coming... please be patient] $\text{not}(A \Rightarrow B)$ ... is equivalent to ... $A \text{ and }(\text{not }B)$ an implication is false when
the hypothesis is true and
the conclusion is false
biconditional statement [beautiful math coming... please be patient] $A \Longleftrightarrow B$ ... is equivalent to ... $(A \Rightarrow B) \text{ and } (B \Rightarrow A)$ this is justification for the ‘double arrow’
that is used for equivalence
TAUTOLOGIES

A tautology is a mathematical sentence form that is always true.
For example, ‘$\,(A\text{ and }B)\Rightarrow A\,$’ is a tautology, as the truth table below confirms:

$A$ $B$ $A\text{ and }B$ $(A\text{ and }B)\Rightarrow A$
T T T T
T F F T
F T F T
F F F T

Here's the intuition for the tautology ‘$\,(A\text{ and } B)\Rightarrow A\,$’:
the only way an ‘and’ sentence is true is when both subsentences are true.
Thus, if ‘$\,A\text{ and }B\,$’ is true, then (in particular) $\,A\,$ must be true.
Check that ‘$\,(A\text{ and }B)\Rightarrow B\,$’ is also a tautology.

Note that a logical equivalence is a particular type of tautology—
one that tells us that two different sentence forms always have the same truth values, regardless of the truth values of the components.

Difference in usage between the words ‘tautology’ and ‘identity’

Although the words ‘identity’ and ‘tautology’ are both used in mathematics to refer to sentences that are always true,
there is a slight difference in usage.

The word ‘tautology’ tends to be used in logic, where the ‘pieces’ of the (always true) sentence have the choice of being either true or false.
For example, ‘$\,(A\text{ and }B)\Rightarrow A\,$’ is a tautology:   the sentence is always true;   the ‘pieces’ $\,A\,$ and $\,B\,$ can be true or false.
The sentence ‘$\,A\text{ or } (\text{not }A)\,$’ is another tautology:   the sentence is always true; the ‘piece’ $\,A\,$ can be true or false.

The word ‘identity’, on the other hand, tends to be used for mathematical sentences outside of the realm of logic that are always true.
Here are some examples from algebra and trigonometry (and don't worry if something is unfamiliar here):

We finish with two important tautologies that allow us to ‘chain’ results together:

LAW OF SYLLOGISM
The following compound mathematical sentence is true,
for all possible truth values of [beautiful math coming... please be patient] $\,P\,$, $\,Q\,$, and $\,R\,$:

If [beautiful math coming... please be patient] $\,((P\Rightarrow Q)\text{ and }(Q\Rightarrow R))\,$ then $\,(P\Rightarrow R)\,$

Here's the intuition:
Whenever [beautiful math coming... please be patient] $\,P\,$ is true, so is $\,Q\,$; and,
whenever $\,Q\,$ is true, so is $\,R\,$; so,
whenever $\,P\,$ is true, so is $\,R\,$.

LAW OF DETACHMENT
The following compound mathematical sentence is true,
for all possible truth values of $\,P\,$ and $\,Q\,$:

[beautiful math coming... please be patient] $(P\text{ and }(P\Rightarrow Q))\ \Rightarrow\ Q$

Here's the intuition:
If $\,P\,$ is true; and,
whenever $\,P\,$ is true, so is $\,Q\,$; then,
$\,Q\,$ must be true.

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Applying Logical Equivalences
to Algebraic and Geometric Statements


On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
(MAX is 16; there are 16 different problem types.)