3 Boolean Operator Diagrams

In all of these figures, a white pixel indicates a 0 value, and a black pixel indicates a 1 value⁸. The images were generated using the AVS network shown in Figure 6, using the ip_logical image processing module. However, this module considers white pixels to be 1 and black pixels to be 0; this is the reason for the ip_logical NOT modules just after reading the input images and just before writing the output images.

Figure 6: The AVS 5 network used to generate Figure 8 from Figure 7.

Figure 7 shows the input images, each contains a single circle at a different offset. The region inside the circle is defined to be 1, the region outside is 0.

Figure 7: The boolean operator input images.

[left] [right]

Figure 8 illustrates the effect of combining Figures 7a and 7b using the boolean operators AND, NAND, OR, NOR, and XOR respectively.

Figure 8: The boolean operator output images, the result of combining Figures 7a and 7b with the indicated logical operators.

[and] [nand] [or] [nor] [xor]

Feature to note are:

• Figure 8a, depicting AND, is black in the region where both of the input images in Figure 7 are black. This is consistent with the definition of the AND operation.
• Figure 8c, depicting OR, is black in the region where either of the input images in Figure 7 are black. This is consistent with the definition of the OR operation.
• Figure 8e, depicting XOR, is black in the regions where only one, but not both, of the input images in Figure 7 are black. This is consistent with the definition of the XOR operation.
• NAND is, by definition, a negated AND, and, similarly, NOR is a negated OR. This is reflected in Figure 8 by these two pairs or images (Figures 8a - 8b, and 8c - 8d) being ``opposite'', in the sense that what is black in one is white in the other, and vice-verso.

These diagrams are also called VENN DIAGRAMS (John Venn, 1834-1923, [1, Section 2.3, Page 13]), and are frequently used as demonstrative aids in elementary boolean logic and set theory (where AND corresponds to set intersection and OR corresponds to set union).