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Example 6: Prove that (a ∧ (a → b)) → ∼a is a contingency.
              Ans.
                      a    b    ∼a   a → b    a ∧ (a → b)  (a ∧ (a → b)) → ∼a
                      0    0    1     1          0               1
                      0    1    1     1          0               1
                      1    0    0     0          0               1
                      1    1    0     1          1               0

              The last column has both 0 and 1 as truth values. So, it is a contingency.

                  2.7 LOGIC GATES
              Logic gates are the digital circuits that depict a logical relationship between the input and output voltages of the
              circuits. They are the building blocks of a digital circuit.







                                       LOGIC           AND        OR          XOR         NOT
                                      GATES



                                                            NAND       NOR        XNOR



              A physical gate acts as a barrier and controls entry into the building. Similarly, logic gates are used to control the flow
              of information based on the logical relations. A logic gate can accept one or more inputs but always produces a single
              output. They produce signal 1 as an output when the input logic requirement is satisfied, otherwise signal 0.
              Logic gates require a power supply. In both input and output, 0 volt represents OFF (0) and 1 volt represents ON (1).
              Each logic gate is depicted with a specific graphical symbol.

              The three basic logical operations namely conjunction (and), disjunction (or) and negation (not) are represented by
              their corresponding logic gates discussed below. These gates are also called the fundamental gates.

              2.7.1 NOT Gate
              The NOT gate is a logical gate that always gives the opposite output of the input signal. NOT gate requires a single input
              and is also called a unary gate. It inverts the output. Input 1 is changed to 0 and 0 to 1. NOT gate is represented as a
                                   –
              complement (') or bar (  ) at the top.
              The symbol and truth table of NOT gate are given below:

                                                                          A           A'
                                         A                      A'
                                                                          0           1
                                                                          1           0
                                           Symbol of NOT gate
                                                                     Truth table of NOT gate
              2.7.2 AND Gate
              The AND gate is a logical gate that always gives single output for two and more than two input signals. The AND gate
              requires at least two inputs and is called a binary gate. It produces a high output (1) only when all the inputs are
              high (1). In all other cases, low output (0) is produced. AND operation is represented by a dot (.).



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