Question 1176997: ~F → G I - F v G
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down this logical expression and determine the relationship between `~F → G` and `F v G`.
**Understanding the Symbols**
* `~`: Negation (NOT)
* `→`: Implication (IF...THEN)
* `v`: Disjunction (OR)
**Breaking Down the Expressions**
1. **`~F → G`** (NOT F implies G)
* This statement is true in all cases except when `~F` is true and `G` is false.
* In other words, it's false only when `F` is false and `G` is false.
2. **`F v G`** (F OR G)
* This statement is true when either `F` is true, `G` is true, or both are true.
* It is only false when both `F` and `G` are false.
**Truth Table**
To see the relationship clearly, let's create a truth table:
| F | G | ~F | ~F → G | F v G |
| :---- | :---- | :---- | :----- | :---- |
| True | True | False | True | True |
| True | False | False | True | True |
| False | True | True | True | True |
| False | False | True | False | False |
**Analysis**
* Notice that the truth values for `~F → G` and `F v G` are identical in all rows of the truth table.
**Conclusion**
Therefore, `~F → G` is logically equivalent to `F v G`. They have the same truth values under all possible combinations of `F` and `G`.
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