Definition:Conjunction/General Definition
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Definition
Let $p_1, p_2, \ldots, p_n$ be statements.
The conjunction of $p_1, p_2, \ldots, p_n$ is defined as:
- $\ds \bigwedge_{i \mathop = 1}^n \ p_i = \begin {cases} p_1 & : n = 1 \\ & \\ \ds \paren {\bigwedge_{i \mathop = 1}^{n - 1} p_i} \land p_n & : n > 1 \end {cases}$
That is:
- $\ds \bigwedge_{i \mathop = 1}^n \ p_i = p_1 \land p_2 \land \cdots \land p_{n - 1} \land p_n$
In terms of the set $P = \set {p_1, \ldots, p_n}$, this can also be rendered:
- $\ds \bigwedge P$
and is referred to as the conjunction of $P$.
Also see
- Conjunction is Associative which validates the construction.
Sources
- 1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 5$: Exercises, Group $\text{III}$
- 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.5.3$: Definition $2.51$