# Minterms and Maxterms **Combinations of Boolean variables in which each variable appears once.** *Minterms* and *maxterms* are combinations of Boolean variables used in the standard expression of [[Boolean Function|Boolean functions]]. A function with $n$ variables will have $2^{n}$ of either minterms or maxterms. Minterms and maxterms are written for all combinations of Boolean variables such that its value is $1$ or $0$, respectively. ## Minterms > [!Minterm] > A *product term* with a truth value of *1* in which each variable appears once, in complemented or uncomplemented form. A variable in a minterm is *complemented* for $0$ and *not complemented* for $1$. | $X$ | $Y$ | Product Term | Symbol | |:---:|:---:|:----------------:|:------:| | $0$ | $0$ | $\bar{X}\bar{Y}$ | $m_0$ | | $0$ | $1$ | $\bar{X}Y$ | $m_1$ | | $1$ | $0$ | $X\bar{Y}$ | $m_2$ | | $1$ | $1$ | $XY$ | $m_3$ | Note that if *any one* minterm is true, *all other* minterms are false. ## Maxterms > [!Maxterm] > A *sum term* with a value of *0* in which each variable appears once, in complemented or uncomplemented form. A variable in a maxterm is *complemented* for $1$ and *not complemented* for $0$. | $X$ | $Y$ | Sum Term | Symbol | |:---:|:---:|:-----------------:|:------:| | $0$ | $0$ | $X+Y$ | $M_0$ | | $0$ | $1$ | $X+\bar{Y}$ | $M_1$ | | $1$ | $0$ | $\bar{X}+Y$ | $M_2$ | | $1$ | $1$ | $\bar{X}+\bar{Y}$ | $M_3$ | Note that if *any one* maxterm is false, *all other* maxterms are true.