The truthfunctions include conjunction, disjunction, and implication.
The words and and so are both grammatical conjunctions joining the sentences (A) and (B) to form the compound sentences (C) and (D). The and in (C) is a logical connective, since the truth of (C) is completely determined by (A) and (B): it would make no sense to affirm (A) and (B) but deny (C). However so in (D) is NOT a logical connective, since it would be quite reasonable to affirm (A) and (B) but deny (D): perhaps, after all, Jill went up the hill fetch a pail of water, not because Jack had gone up the Hill at all. Thus and is a logical connective but so is not. In the realm of pure logic, (C) is a compound statement but (D) is not. (D) cannot be broken into parts using only the logic of statements, the realm of cause and effect being proper to science rather than logic.
Various English words and word pairs express truthfunctions, and some of them are synonymous. Examples (with the name of the relationship in parentheses) are:
"and" (conjunction), "or" (inclusive or exclusive disjunction), "implies" (implication), "if...then" (implication), "if and only if" (equivalence), "only if" (implication), "just in case" (equivalence), "but" (conjunction), "however" (conjunction) , "not both" (NAND), "neither...nor" (NOR). The word "not" (negation) and "it is false that" (negation) "it is not the case that" (negation) are also English words expressing a logical connective, even though they are applied to a single statement, and do not connect two statements.
Logical connectives can be used to link more than two statements. A more technical definition is that an "nary logical connective" is a function which assigns truth values "true" or "false" to ntuples of truth values.
The basic logical operators are:

Some others are:

For example, the statements it is raining and I am indoors can be reformed using various different connectives to form sentences that relate the two in ways which augment their meaning:
If we write 'P' for It is raining and 'Q' for I am indoors and we use the usual symbols for logical connectives, then the above examples could be represented in symbols, respectively:
There are sixteen different Boolean functions, associating the inputs P and Q with four digit binary outputs.
The following table shows important equivalences like the De Morgan's laws (lines 1000 and 1110) or the law of Contraposition (line 1101).
The edges in the Hasse diagram to the right can be seen as implication arrows pointing upwards. That means, functions represented by lower knots imply functions represented by higher knots, when the knots are connected by an edge.
These imagemap graphics contain links and additional information. Move your mouse horizontally over the four bit truth value outputs in the table, to see them explained.
Not all of these operators are necessary for a functionally complete logical calculus. Certain compound statements are logically equivalent. For example, ¬P ∨ Q is logically equivalent to P → Q;. So the conditional operator "→" is not necessary if you have "¬" (not) and "∨" (or).
The smallest set of operators which still expresses every statement which is expressible in the propositional calculus is called a minimal functionally complete set. A minimally complete set of operators is achieved by NAND alone { ↓ } and NOR alone { ↑ }.
The following are the functionally complete sets (of cardinality not exceeding 2) of operators whose arities do not exceed 2:
{ ↓ }, { ↑ }, { $rightarrow$, $neg$ }, { $rightarrow$, $notleftrightarrow$ }, { $neg$, ⊂ }, { $rightarrow$, ⊄ }, { $vee$, $neg$ }, { $rightarrow$, ⊅ }, { ⊄, $neg$ }, { ⊂, $notleftrightarrow$ }, { ⊅, $neg$ }, { ⊂, ⊄ }, { $wedge$, $neg$ }, { ⊂, ⊅ }, { $bot$, $rightarrow$ }, { ⊄, $leftrightarrow$ }, { ⊅, $leftrightarrow$ }
A set of operators is functionally complete if and only if for each of the following five properties it contains at least one member lacking it:
In twovalued logic there are 2 nullary operators (constants), 4 unary operators, 16 binary operators, 256 ternary operators, and $2^\{2^n\}$ nary operators. In three valued logic there are 3 nullary operators (constants), 27 unary operators, 19683 binary operators, 7625597484987 ternary operators, and $3^\{3^n\}$ nary operators. An nary operator in kvalued logic is a function from $mathbb\{Z\}\_k^n\; to\; mathbb\{Z\}\_k$. Therefore the number of such operators is $mathbb\{Z\}\_k^$
However, some of the operators of a particular arity are actually degenerate forms that perform a lowerarity operation on some of the inputs and ignores the rest of the inputs. Out of the 256 ternary boolean operators cited above, $binom\{3\}\{2\}cdot\; 16\; \; binom\{3\}\{1\}cdot\; 4\; +\; binom\{3\}\{0\}cdot\; 2$ of them are such degenerate forms of binary or lowerarity operators, using the inclusionexclusion principle. The ternary operator $f(x,y,z)=lnot\; x$ is one such operator which is actually a unary operator applied to one input, and ignoring the other two inputs.
"Not" is a unary operator, it takes a single term (¬P). The rest are binary operators, taking two terms to make a compound statement (P $wedge$ Q, P $vee$ Q, P → Q, P ↔ Q).
The set of logical operators $Omega!$ may be partitioned into disjoint subsets as follows:
In this partition, $Omega\_j!$ is the set of operator symbols of arity $j!$.
In the more familiar propositional calculi, $Omega!$ is typically partitioned as follows:
Here is a table that shows a commonly used precedence of logical operators.
Operator  Precedence 

¬  1 
∧  2 
∨  3 
→  4 
5 
The order of precedence determines which connective is the "main connective" when interpreting a nonatomic formula.
The "logical equivalence" of "NAND alone", "NOR alone", and "NOT and AND" is similar to Turing equivalence.
Is some new technology (such as reversible computing, clockless logic, or quantum dots computing) "functionally complete", in that it can be used to build computers that can do all the sorts of computation that CMOSbased computers can do? If it can implement the NAND operator, only then is it functionally complete.
That fact that all logical connectives can be expressed with NOR alone is demonstrated by the Apollo guidance computer.