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Which of the two following propositions are equivalent in the sense that one can always be substituted for the other one in any

proposition without changing its truth value?

1. first proposition $:\text{P} \Rightarrow \text{Q};$ second proposition $:\neg \text{P} \vee \text{Q}$
2. first proposition $:\neg \text{P};$ second proposition $:\text{P} \Rightarrow \text{False}$
3. first proposition $:\neg \text{P};$ second proposition $:\text{False} \Rightarrow \text{P}$
4. first proposition $:\neg \text{P};$ second proposition $:\neg \text{P} \vee \text{Q}$

1. first proposition $: \text{P} \Rightarrow \text{Q}$ second proposition $: \neg \text{P} \vee \text{Q}$

$\text{Answer}$: yes

$\text{Example reasoning}$:

All rows in the truth table evaluate to the same truth value.

$$\begin{array} {|c|c|c|c|} \hline \text{P} & \text{Q} & \text{P} \Rightarrow \text{Q} & \neg \text{P} \vee \text{Q} \\\hline \text{T} & \text{T} & \text{T} & \text{T} \\\hline \text{T} & \text{F} & \text{F} & \text{F} \\\hline \text{F} & \text{T} & \text{T} & \text{T} \\\hline \text{F} & \text{F} & \text{T} & \text{T} \\\hline \end{array}$$

1. first proposition $: \neg \text{P}$ second proposition $: \text{P} \Rightarrow \text{False}$

$\text{Answer}:$ yes

1. first proposition $: \neg \text{P}$ second proposition $: \text{False} \Rightarrow \text{P}$

$\text{Answer}:$ no

1. first proposition $: \neg \text{P}$ second proposition $: \neg \text{P} \vee \text{Q}$

$\text{Answer}:$ no

Detailed Video Solution: https://youtu.be/nclBhBmtz2g?t=548