Definition:Conjunction
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Definition
Conjunction is a binary connective written symbolically as $p \land q$ whose behaviour is as follows:
- $p \land q$
is defined as:
- $p$ is true and $q$ is true.
This is called the conjunction of $p$ and $q$.
The statements $p$ and $q$ are known as the conjuncts.
$p \land q$ is voiced:
- $p$ and $q$.
Boolean Interpretation
From the above, we see that the boolean interpretations for $\mathbf A \land \mathbf B$ under the model $\mathcal M$ are:
- $\left({\mathbf A \land \mathbf B}\right)_{\mathcal M} = \begin{cases} T & : \mathbf A_{\mathcal M} = T \text{ and } \mathbf B_{\mathcal M} = T \\ F & : \text {otherwise} \end{cases}$
Generalized Notation
- $\displaystyle \bigwedge_{i=1}^n \ p_i = p_1 \land p_2 \land \cdots \land p_{n-1} \land p_n$
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Complement
The complement of $\land$ is the NAND operator.
Truth Function
The conjunction connective defines the truth function $f^\land$ as follows:
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle f^\land \left({F, F}\right)\) | \(=\) | \(\displaystyle F\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle f^\land \left({F, T}\right)\) | \(=\) | \(\displaystyle F\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle f^\land \left({T, F}\right)\) | \(=\) | \(\displaystyle F\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle f^\land \left({T, T}\right)\) | \(=\) | \(\displaystyle T\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) |
Truth Table
The truth table of $p \land q$ and its complement is as follows:
$\begin{array}{|cc||c|c|} \hline p & q & p \land q & p \uparrow q \\ \hline F & F & F & T \\ F & T & F & T \\ T & F & F & T \\ T & T & T & F \\ \hline \end{array}$
Semantics of the Conjunction
The conjunction is used to symbolise any statement in natural language such that two substatements are held to be true simultaneously.
Thus it is also used to symbolise the concept of but.
Thus it can be also interpreted as:
- $p$ but $q$
- $p$; however, $q$
- $p$; on the other hand $q$
- Not only $p$ but also $q$
- Despite $p$, $q$
Notational Variants
Various symbols are encountered that denote the concept of conjunction:
| Symbol | Origin | Known as |
|---|---|---|
| $p \land q$ | wedge | |
| $p\ \mathsf{AND} \ q$ | ||
| $p \ . \ q$ | Bertrand Russell and Alfred North Whitehead: Principia Mathematica (1910) | dot |
| $p \ \And \ q$ | Ampersand | |
| $\operatorname K p q$ | Łukasiewicz's Polish notation |
Sources
- Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 2$: The Axiom of Specification
- Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning (1964): $\text{II}: \S 1$
- E.J. Lemmon: Beginning Logic (1965): $\S 1.3$
- Alan G. Hamilton: Logic for Mathematicians (1978): $\S 1.2$
- D.J. O'Connor and Betty Powell: Elementary Logic (1980): $\S 1.2, \ \S 1.3$ and Appendix
- Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.1$
- H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): $\S 1.1$
- Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems (2000): $\S 1.1, \ \S 1.4$ Fig. $1.8, \ 1.9$
