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Copy file name to clipboardexpand all lines: docs/binary-algebra/boolean-functions.md
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## Mathematical definition
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If $k$ is the number of Boolean variables of the function, then the function $f(x_1,\ldots,x_k)$ and its domain and codomain are defined as $f:\\{0,1\\}^k\rightarrow\\{0,1\\}$
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If $k$ is the number of Boolean variables of the function, then the function $f(x_1,\ldots,x_k)$ and its domain and codomain are defined as $f:\{0,1\}^k\rightarrow\{0,1\}$
More details can be found in Section 1.9 "Shannon's Expansion Theorem" in {% cite donzellini2018introduction --file books %} and in Section 3.2 "Switching functions" in {% cite kohavi2010switching --file books %}.
Copy file name to clipboardexpand all lines: docs/binary-representation/negative-quantities.md
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## Signed and unsigned numbers
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Currently, we have just looked at **unsigned** numbers - they can only be positive, as there is no sign. However, sometimes we need to work with **negative numbers** too. To do this, we add a **sign bit** on the far left of the binary number, which indicates whether the number is positive (\`0\`) or negative(\`1\`).
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Currently, we have just looked at **unsigned** numbers - they can only be positive, as there is no sign. However, sometimes we need to work with **negative numbers** too. To do this, we add a **sign bit** on the far left of the binary number, which indicates whether the number is positive (`0`) or negative(`1`).
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For example, the number \`10000011\` would be \`131\` if the number is **unsigned**, but if the number is **signed**, the actual representation would be \`-3\`
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For example, the number `10000011` would be `131` if the number is **unsigned**, but if the number is **signed**, the actual representation would be `-3`
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- The first bit \`1\` represents that the number is negative
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- The remaining bits \`0000011\` represent the actual number, \`3\`
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- The first bit `1` represents that the number is negative
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- The remaining bits `0000011` represent the actual number, `3`
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The downside to using a signed number is that it removes one bit from the actual number representation, halving the maximum value.
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- The minimum and maximum values for an \`unsigned 8-bit\` number would be \`0\` to \`2<sup>8</sup>-1\` (\`0\` to \`255\`)
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- The minimum and maximum values for a \`signed 8-bit\` number would be \`-2<sup>7</sup>-1\` to \`2<sup>7</sup>-1\` (\`-127\` to \`127\`)
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- The minimum and maximum values for an `unsigned 8-bit` number would be `0` to `2<sup>8</sup>-1` (`0` to `255`)
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- The minimum and maximum values for a `signed 8-bit` number would be `-2<sup>7</sup>-1` to `2<sup>7</sup>-1` (`-127` to `127`)
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## Complements in binary system
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{:.quiz}
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1. Using signed two's complement notation,what is the decimal value of 00010001 ?
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1. Using signed two's complement notation,what is the decimal value of 00010001 ?
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* -47
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* -149
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1. +17
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2. Using signed two's complement notation,what is the decimal value of 00010001 ?
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2. Using signed two's complement notation,what is the decimal value of 00010001 ?
Copy file name to clipboardexpand all lines: docs/comb-lsi/alu.md
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Now you can take up the 1 bit ALU as block and construct a 4 bit ALU, which performs all the functions of the 1 bit ALU on the 4 bit inputs. Thus a single building block can be constructed and used recursively. The inputs A and B are four bits and the output is 4 bit as well. Figure below illustrates it:
{% include image.html url='/assets/images/360px-4BITALU.jpg' description='4 bit ALU' %}
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There are a few important takeaways here:
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- The selection lines MO and M1 select the function ALU performs. These selection lines combined with the input arguments and desired functions, an Instruction Set can be formed.
Copy file name to clipboardexpand all lines: docs/logic-design/functional-description.md
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Combinational circuits consist of Logic gates. These circuits operate with binary values. The output(s) of combinational circuit depends on the combination of present inputs. The following figure shows the **block diagram** of combinational circuit.
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