This chapter presents the similarities between C# and C++; some differences (including global variables and functions; the preprocessor, compilation, namespaces; printing), concluding with classes, templates, pointers memory allocation and deallocation. | (CSC 102) Lecture 4 Discrete Structures Previous Lecture Summary Conditional Propositions. Negation, Inverse and Converse of the conditional statements. Contraposition Bi-conditional statements. Necessary and Sufficient Conditions. Conditional statements and their Logical equivalences. 3 Applications of Logic Lectures outline Basic Logic gates Circuits using logic gates Boolean Algebra Adders Reductions of circuits Basic Logic Gates Not where x = ¬x And where xy = x y Or where x+y = x y Nand where ¬(xy)= xy Nor Xor Constructing Circuits Here is the circuit of the statement (p q) (~p q) (p ~q) 6 Cont. Following is the circuit output of the following statement (x + y) ¬ y Designing a circuitt for a given input/output Here is the out put we can write it as following Designing a circuitt for a given input/output Here is the circuit of the previous input/output Boolean Algebra Just like Boolean logic, variables can only be 1 or 0, instead of true/false Not ~0 = 1 ~1 = 0 Or is used as a plus And is used as a multiplication 0+0 = 0 0 * 0 = 0 0+1=1 0 * 1 = 0 1+0=1 1 * 0 = 0 1+1= ? 1 * 1 = 1 Half Adder Consider adding two 1-bit binary numbers x and y 0+0 = 0 0+1 = 1 1+0 = 1 1+1 = 10 Carry is x AND y Sum is x XOR y The circuit to compute this is called a half-adder. Circuit of Half Adder Sum = x XOR y Carry = x AND y Using Half adders We can then use a half-adder to compute the sum of two Boolean numbers 1 1 0 0 + 1 1 1 0 0 1 0 ? 0 0 1 How to fix that We need to create an adder that can take a carry bit as an additional input Inputs: x, y, carry in Outputs: sum, carry out This is called a full adder Will add x and y with a half-adder Will add the sum of that to the carry in What about the carry out? It’s 1 if either (or both): x+y = 10 x+y = 01 and carry in = 1 The Full adder The “HA” boxes are half-adders The Full adder The full circuitry of the full adder Logical Expression Following is the circuit representations of the statement Cont . The above . | (CSC 102) Lecture 4 Discrete Structures Previous Lecture Summary Conditional Propositions. Negation, Inverse and Converse of the conditional statements. Contraposition Bi-conditional statements. Necessary and Sufficient Conditions. Conditional statements and their Logical equivalences. 3 Applications of Logic Lectures outline Basic Logic gates Circuits using logic gates Boolean Algebra Adders Reductions of circuits Basic Logic Gates Not where x = ¬x And where xy = x y Or where x+y = x y Nand where ¬(xy)= xy Nor Xor Constructing Circuits Here is the circuit of the statement (p q) (~p q) (p ~q) 6 Cont. Following is the circuit output of the following statement (x + y) ¬ y Designing a circuitt for a given input/output Here is the out put we can write it as following Designing a circuitt for a given input/output Here is the circuit of the previous input/output Boolean Algebra Just like Boolean logic, variables can only be 1 or 0, instead of true/false Not ~0 = 1 ~1 = 0 .
