Material Biconditional

The material biconditional or XNOR of two operands is true when they have the same value — both $0$ or both $1$. It’s the complement of XOR.

$X$	$Y$	$X\leftrightarrow Y$
0	0	1
0	1	0
1	0	0
1	1	1

In SOP form, $X\odot Y=XY+\overline{X}\overline{Y}$ — true when both are $1$ or both are $0$.

XNOR is also called logical equality because it’s exactly $X=Y$ in Boolean terms. That’s why an XNOR gate is the natural building block for an equality comparator: bit-wise XNOR each pair, then AND all the results — output is $1$ only when every bit matches.

Notation

In electronics: $\odot$ (or sometimes $\overline{\oplus }$). In formal logic: $\leftrightarrow$ (the biconditional arrow).

The corresponding hardware element is the XNOR gate — drawn as an XOR with a bubble on the output.

In context

A useful identity that comes up in MUX-based logic:

$x_2\odot x_3=\overline{x_2\oplus x_3}$

If a 2-to-1 multiplexer has $\overline{x_3}$ and $x_3$ as data inputs (with $\overline{x_3}$ on the $0$ line and $x_3$ on the $1$ line) and $x_2$ as the select, the output is $\overline{x_2}\overline{x_3}+x_2{x}_3=x_2\odot x_3$.

Like XOR, XNOR is commutative and associative. $X\odot 1=X$ but $X\odot 0=\overline{X}$ — XNOR with $0$ inverts.