Pairs

The karnaugh map uses Pairs, Quads and Octets to get simplified logic circuits.

Pair

A pair is used to drop one variable and its complement to get simplified booleon equation.

Let us consider a Karnaugh map (Fig.2.19) which contains a pair of 1s that are horizontally adjacent and a pair of 1s that are vertically adjacent. Therefore the karnaugh map contains two pairs.

Horizontal pair

i) In the horizontal pair, the first 1 represents the product BD

ii) The second 1 represents the product BCD

The sum of products

Explanation

When we move from the first 1 to the second 1 in the horizontal direction, only one variable goes from complemented form to uncomplemented form i.e.,  to C.

The other variables (, B and D) don't change form.

Whenever this happen, we can eliminate the variable that changes form i.e,  and C.

∴ In horizontal pair, we can directly write the SOP equation by dropping a variable and its complement i.e,  and C.

The equation is Y = BD,

Vertical Pair

(i) In the vertical pair, the first 1 corresponds to the product 

(ii) The second 1 corresponds to the product .

∴  The sum of product equation is

Explanation

In vertical pair, we can directly write the equation by dropping one variable and its complement.

Here in the vertical direction one variable changes from uncomplemented form to complemented form i.e.,B to .

All other variables, (A,,) retain their original form. Therefore B and  shall be dropped out.

Since two pairs exist in the karnaugh map, the corresponding boolean equation for this map is