To analyze the given pair of statements:
1. [tex]$D \cdot \sim R$[/tex].
2. [tex]$R \cdot \sim D$[/tex].
Let's carefully break down each statement:
1. [tex]$D \cdot \sim R$[/tex] means "D and not R".
2. [tex]$R \cdot \sim D$[/tex] means "R and not D".
### Step-by-Step Analysis
#### Examining [tex]$D \cdot \sim R$[/tex]
- This statement means that D is true, and R is false simultaneously.
- Symbolically, [tex]\( D = \text{True} \)[/tex] and [tex]\( R = \text{False} \)[/tex].
#### Examining [tex]$R \cdot \sim D$[/tex]
- This statement means that R is true, and D is false simultaneously.
- Symbolically, [tex]\( R = \text{True} \)[/tex] and [tex]\( D = \text{False} \)[/tex].
### Logical Implications
- For the first statement to hold true, D must be true, and R must be false.
- For the second statement to hold true, R must be true, and D must be false.
### Contradictions and Consistency
- If [tex]\( D \)[/tex] is true, then [tex]\( R \)[/tex] must be false for the first statement.
- If [tex]\( R \)[/tex] is true, then [tex]\( D \)[/tex] must be false for the second statement.
- These two requirements ([tex]\( D = \text{True}, R = \text{False} \)[/tex] and [tex]\( R = \text{True}, D = \text{False} \)[/tex]) cannot both be true at the same time.
### Conclusion
- Since the statements [tex]$D \cdot \sim R$[/tex] and [tex]$R \cdot \sim D$[/tex] both cannot be true simultaneously, they are logically inconsistent with each other.
Therefore, the answer to the question is:
c) Inconsistent.
