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Statements, Negation, Conjunction, and Disjunction

Discrete Mathematics Truth Tables Statements, Negation, Conjunction, and Disjunction

(Mathematical) Statement

A (mathematical) statement or proposition is a sentence that is true or false, but not both.

For example, “One of my cats is grey and white in color” or “My brother is 22 years old” are statements (albeit not very mathematically precise). A sentence such as “$x+y=0$” is not a statement because it is possible for the sentence to be true for certain values of $x$ and $y$ (such as if $x=2$ and $y=-2$) and false for others (such as if $x=10$ and $y=3$). So, in short, a sentence is a statement, essentially if it is unconditionally true or unconditionally false (i.e. either true or false “no matter what”).

Mathematics, as a whole, is a collection of true statements logically derived from a set of core assumptions (called axioms) and definitions. New compound statements can be formed by combining statements in a few different ways using what are called logical connectives.

Negation

Let $p$ be a statement. The negation of $p$, written $\neg p$ and pronounced either “not $p$” is the statement that has the opposite truth value of $p$.

That is, $\neg p$ is true when $p$ is a false statement, and $\neg p$ is false if $p$ is a true statement. This is represented in the truth table for $\neg p$ below.

$p$	$\neg p$
False	True
True	False

Usually, instead of writing “true” and “false” we just use “T” and “F”

Examples of Negation

$p$	$\neg p$
“Emily is a friendly person.”	“Emily is NOT a friendly person
“It will not rain today”	“It will rain today”
“$2<6$”	“$2\geq 6$”

Conjunction

Let $p$ and $q$ be statements. The conjunction of $p$ and $q$, written $p\wedge q$ and pronounced “$p$ AND $q$,” is the statement that is true provided that $p$ and $q$ are simultaneously true. If either $p$ or $q$ is false, then $p\wedge q$ is defined to be false.

For the conjunction $p\wedge q$ of $p$ and $q$ to be true, it is required that both $p$ and $q$ are true. This is the only way that $p\wedge q$ is true. This can be seen in the truth table below.

$p$	$q$	$p\wedge q$
F	F	F
F	T	F
T	F	F
T	T	T

Note that only in the last row, where p and q are true is the conjunction true.

Exandples of Conjunction

$p$	$q$	$p\wedge q$
“It is raining”	“I am dry”	“It is raining and I am dry”
“I hate working”	“I like money”	“I hate working but (and) I like money”
“$1\leq 5$”	“$5\leq 10$”	“$1\leq 5$ and $5\leq 10$” or rather “$1\leq 5\leq 10$”

“But” Wait!

Note that the word “but” has the same meaning as the word “and” from a logical standpoint. So, statements like “It is cold but it is not snowing” is logically equivalent to “It is cold and it is not snowing.” The word “but” is used when two statements are true, but it seems that the statements oppose one another in content (such as this sentence, for example!). There is no rule for when to use “but” versus “and.” Just use whichever sounds right intuitively; either one is correct because “and” and “but” are logically treated as the same. Similar remarks can be made for the word “while” such as in the statement “My cats eat steak while I eat cereal.”

A Note on “Neither… Nor…”

Let $p$ and $q$ be statements. The statement “Neither $p$ nor $q$” is logically equivalent to “$\neg p$ and $\neg q$.” For example, the statement “The glass is neither full nor empty” is logically the same as saying “The glass is not full and the glass is not empty.”

Disjunction

Let $p$ and $q$ be statements. The disjunction of $p$ and $q$, written $p\vee q$ and pronounced $p$ OR $q$, is the statement that is true when $p$ is true, $q$ is true, or both $p$ and $q$ are true, and is false when both $p$ and $q$ are false.

Logical “OR” is slightly different from our intuitive idea of “OR.” In common parlance, “OR” implies exclusivity; as in “either $p$ or $q$ (but not both).” Logical “OR” allows for both $p$ and $q$ to be true. All this is expressed in the truth table below.

$p$	$q$	$p\vee q$
F	F	F
F	T	T
T	F	T
T	T	T

Note the only way $p\vee q$ is false is if both statements $p$ and $q$ are false.

Examples of Disjunction

$p$	$q$	$p\vee q$
“I’ll eat cake”	“I’ll eat ice cream”	“I’ll eat cake or I’ll eat ice cream”
“I prefer blue”	“I prefer red”	“I prefer blue or I prefer red”
“$1<0$”	“$5=4$”	“$1<0$ or $5=4$”
“$x=2$”	“$x=-2$”	“$x=\pm 2$”

The symbols $\vee$, $\wedge$, and $\neg$ above, for OR, AND, and NOT, respectively, are examples of what are called logical connectives.

Tautology and Contradiction

A statement $t$ that is always true, regardless of the truth values of individual statements that may comprise $t$, is called a tautology.

A statement $c$ that is always false, regardless of the truth values of individual statements that may comprise $c$, is called a contradiction.

A tautology is a statement that is true no matter what. A contradiction is a statement that is false no matter what. For example, the statement “The fish is green OR the fish is not green” is a tautology; the statement is true no matter what color the fish is. Contrarily, the statement “x>1 AND x<-5” is a contradiction; no matter what value $x$ takes (no number is both greater than 1 and less than -5).

$p\wedge q$
$p\vee q$
$\neg p \wedge q$
$\neg p \vee \neg q$
$\neg p \wedge p$
$ q \vee \neg q$

“The fish is green and the fish is black”
“The fish is green or the fish is black”
“The fish is not green and the fish is black”
“The fish is not green or the fish is not black”
“The fish is not green and the fish is green”
“The fish is black or the fish is not black”

Remember: $\neg$ means “not,” $\vee$ means “OR” and $\wedge$ means “AND.”

“$n$ is even and divisible by $3$”
“$n$ is not even and $n$ is divisible by $3$”
“$n$ is even or $n$ is not divisible by $3$”
“$n$ is even or $n$ is not even”

$p\wedge q$
$\neg p \wedge q$
$p \vee \neg q$
$p \vee \neg p$.

$3\in A$ or $0\in B$
$3 \in A$ and $9\in B$
$5\in B$ and $5\notin A$
$1\notin A$ or $1 \notin B$
$1\in A$ or $1 \notin A$
$2\in B$ and $2\notin B$

True
False
False
True
True
False

Recall $A=\{0,1,…,10\}$ and $B=\{3,4,5,6\}$.

$3\in A$ or $0\in B$
- $3\in A$ is true, so the whole statement is true (only need one of the two statements the be true for the entire disjunction to be true.
$3 \in A$ and $9\in B$
- $3 \in A$ is true, but $9\in B$ is false, so the whole statement is false. Both $3\in A$ and $9\in B$ must be true for the whole conjunction to be true.
$5\in B$ and $5\notin A$
- $5\notin A$ is false, so the conjunction is false
$1\notin A$ or $1 \notin B$
- $1 \notin B$ is true, so the whole disjunction is true
$1\in A$ or $1 \notin A$
- Note that these are opposites. Naturally one of them must be true, so the whole disjunction is true.
$2\in B$ and $2\notin B$
1. Since these are opposites, they both cannot be true, so the whole conjunction is false.

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\(p\)	\(q\)	\(p\vee q\)
“I’ll eat cake”	“I’ll eat ice cream”	“I’ll eat cake or I’ll eat ice cream”
“I prefer blue”	“I prefer red”	“I prefer blue or I prefer red”
“\(1<0\)”	“\(5=4\)”	“\(1<0\) or \(5=4\)”
“\(x=2\)”	“\(x=-2\)”	“\(x=\pm 2\)”

\(p\)	\(q\)	\(p\wedge q\)
F	F	F
F	T	F
T	F	F
T	T	T