This is logic involving truth tables.
How can i prove
C¡úD
B¡ÅC
A¡Ä¡×B
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¡àD
To prove the expression ¡àD using truth tables, you need to create a truth table that includes all the variables involved in the expression and evaluate the expression for every possible combination of truth values for those variables.
Here's how you can construct the truth table:
1. Identify the variables involved: A, B, C, and D.
2. Determine the number of rows in the truth table. Since there are 4 variables, the truth table will have 2^4 = 16 rows.
3. Create columns for each variable and the expression ¡àD.
4. In each column for the variables, divide the rows in half and alternate assigning truth values (T) and false values (F) to each variable. This ensures that all possible combinations are represented.
5. Evaluate the expression for each combination of truth values and fill in the corresponding value in the ¡àD column.
6. Fill in the remaining columns for B ¡Å C, C ¡ú D, and A ¡Ä ¡× B using the truth table.
Once the truth table is complete, look at the column for the ¡àD expression. If every row in that column evaluates to true (T), then you have proven that the expression ¡àD is true in all possible cases. If there is at least one row where ¡àD is false (F), then the expression is not always true.
Here's an example of a completed truth table:
A | B | C | D | B ¡Å C | C ¡ú D | A ¡Ä ¡× B | ¡àD |
-----------------------------------------------------
F | F | F | F | ? | ? | ? | ? |
F | F | F | T | ? | ? | ? | ? |
F | F | T | F | ? | ? | ? | ? |
F | F | T | T | ? | ? | ? | ? |
F | T | F | F | ? | ? | ? | ? |
F | T | F | T | ? | ? | ? | ? |
F | T | T | F | ? | ? | ? | ? |
F | T | T | T | ? | ? | ? | ? |
T | F | F | F | ? | ? | ? | ? |
T | F | F | T | ? | ? | ? | ? |
T | F | T | F | ? | ? | ? | ? |
T | F | T | T | ? | ? | ? | ? |
T | T | F | F | ? | ? | ? | ? |
T | T | F | T | ? | ? | ? | ? |
T | T | T | F | ? | ? | ? | ? |
T | T | T | T | ? | ? | ? | ? |
Fill in the remaining columns for B ¡Å C, C ¡ú D, and A ¡Ä ¡× B using the appropriate logical operations. Then, evaluate the ¡àD expression for each row based on the values in the other columns. Replace the question marks (?) with the corresponding results.
Once you have completed the truth table, you can determine if the expression ¡àD is true in all cases or if there are specific combinations of truth values where it evaluates to false.