Logical operators

Negation (not) [$\neg p$]:

let $p$ be a proposition. Then, the negation of $p$ is $\neg p$ and means the statement “It is not the case $p$.” This means that the statement $\neg p$ means “not p.”

Therefore, the truth value of $\neg p$ is the opposite of the truth value for $p$

Example: $p$: “Austin is in Central Texas” $\neg p$: “It is not the case that Austin is in Central Texas”

Example 2: $p$: $8-5=2$ $\neg p$: $8-5\ne 2$

Conjunction (and) [$p\wedge q$]

If we let $p$ and $q$ be propositions, then the conjunction of $p$ and $q$ is denoted by $p\wedge q$.

Read as ”$p$ and $q$”

The truth value of $p\wedge q$ is true only when both $p$ and $q$ are true, and false otherwise.

Example: $p$: The sun is shining $q$: It is raining

$p\wedge q$: The sun is shining and it is raining.

“The sun is shining, but it is raining.”

Disjunction (or) [$p\vee q$]

Let $p$ and $q$ be propositions. The disjunction of $p$ and $q$ is denoted by $p\vee q$.

Read as $p$ or $q$

Evaluates as false when both are false, otherwise is true.

Example: $p$: “I will eat a hamburger” $q$: “I will eat a slice of pizza”

$p\vee q$: “I will eat a hamburger or I will eat a slice of pizza”

“I will eat either a hamburger or a slice of pizza”

Implication (implies) [$p⟹q$]

Note: does not mean “suggests.” Note: does not mean “causes.”

Let $p$ and $q$ be propositions. The implication of $p$ and $q$ is denoted by $p⟹q$.

Read as $p$ implies $q$

Think of it as “if I know this one thing, $p$ does it mean I know this other thing, $q$”

Better put as “if $p$ then $q$”

Statements with it must always have a truth value.

Only false when antecedent is true yet consequent is false, otherwise true.

Truth Table:

p	q	$p⟹q$
T	T	T
T	F	F
F	T	T
F	F	T

$p$ is the antecedent, and $q$ is the consequent.

Example: If it’s summer in Austin, it will be hot.

Example 2: True or False: $(\sqrt{4}=2\wedge \sqrt{9}=4)⟹\sqrt{30}=5$

Antecedent must be true and consequent must be false.

Antecedent is $\sqrt{4}=2\wedge \sqrt{9}=4$, or “the square root of 4 is 2 and the square root of 9 is 4.” This expression is false, thus the antecedent is false.

Consequently, the consequent (lol) does not matter, and the statement is true.

Biconditional (if and only if) [$p\leftrightarrow q$]

Let $p$ and $q$ be propositions. The biconditional statement of $p$ and $q$ is denoted by $p\leftrightarrow q$.

Read as $p$ if and only if $q$

Shorthand: $p\mathrm{iff}q$

The truth value of $p\leftrightarrow q$ is true only when $p$ and $q$ have the same truth values, false otherwise.

p	q	$p\leftrightarrow q$
T	T	T
T	F	F
F	T	F
F	F	T

Example: “You can take the flight if and only if you have a ticket”