Basic Logic Connectives

For basic and important logic connectives are introduce here.

Conjunction

When we have ${\displaystyle p\wedge q}$, we have the conjunction of ${\displaystyle p}$ and ${\displaystyle q}$.

The value of ${\displaystyle p\wedge q}$ is true only when both ${\displaystyle p}$ and ${\displaystyle q}$ are TRUE.

Disjunction

When we have ${\displaystyle p\vee q}$, we have the disjunction of ${\displaystyle p}$ and ${\displaystyle q}$. It is actually the proposition "${\displaystyle p}$ or ${\displaystyle q}$".

Then value of ${\displaystyle p\vee q}$ is false only when both ${\displaystyle p}$ and ${\displaystyle q}$ are FALSE.
It means that if there is at least one proposition of the disjunction is True, then the value of disjunction is TRUE.

Inclusive OR and Exclusive OR

These are two kinds of OR in practical cases. The inclusive OR is actually the disjunction.

The exclusive or, namely XOR, is the proposition which is true only when one of ${\displaystyle p}$ or ${\displaystyle q}$ is true. When both ${\displaystyle p}$ and ${\displaystyle q}$ are true, the XOR is false.

Noted by ${\displaystyle p\oplus q}$, we sometimes need the XOR in discrete math.

Conditional Statement

The conditional statement ${\displaystyle p\to q}$ is the proposition "if ${\displaystyle p}$, then ${\displaystyle q}$".

The conditional statement is false only when ${\displaystyle p}$ is TRUE but ${\displaystyle q}$ is FALSE.
Otherwise:

• When ${\displaystyle p}$ is true(false) and ${\displaystyle q}$ is also true(false), the proposition is obviously true.
• When ${\displaystyle p}$ is false and ${\displaystyle q}$ is true, it means that no mater what ${\displaystyle p}$ is, the proposition ${\displaystyle q}$ is always true.

Biconditional Statement

The biconditional statement ${\displaystyle p\leftrightarrow q}$ is the proposition "${\displaystyle p}$ of and only if ${\displaystyle q}$".

When ${\displaystyle p}$ is true(false) and ${\displaystyle q}$ is also true(false), the proposition is obviously true.

But when the truth value of ${\displaystyle p}$ and ${\displaystyle q}$ are not the same, the proposition is always FALSE.