(related to Proposition: Associativity of Conjunction)

- Let $x,y,z$ be propositions and let $I$ be an interpretation with the valuation function $[[]]_I$.
- Based on the truth table of conjunction, we can construct a truth table for $(x\wedge y) \wedge z$ and $x\wedge (y \wedge z)$ independently for all possible semantics of $x,y,z$:

$[[x]]_I$ | $[[y]]_I$ | $[[z]]_I$ | $[[x \wedge y]]_I$ | $[[y \wedge z]]_I$ | $[[(x \wedge y)\wedge z]]_I$ | $[[x\wedge (y \wedge z)]]_I$ |
---|---|---|---|---|---|---|

$1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ |

$0$ | $1$ | $1$ | $0$ | $1$ | $0$ | $0$ |

$1$ | $0$ | $1$ | $0$ | $0$ | $0$ | $0$ |

$0$ | $0$ | $1$ | $0$ | $0$ | $0$ | $0$ |

$1$ | $1$ | $0$ | $1$ | $0$ | $0$ | $0$ |

$0$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ |

$1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |

$0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |

- Since the values in both columns to the right of the table are equal, it follows that the conjunction operation is associative, i.e. $(x\wedge y) \wedge z =x\wedge (y \wedge z)$.∎

**Mendelson Elliott**: "Theory and Problems of Boolean Algebra and Switching Circuits", McGraw-Hill Book Company, 1982