Consider the following statement:
“We shall go either for a picnic or for trekking”.
Which of the following, if true, would falsify this claim ?

The correct option is "We do not go either for picnic or for trekking."

Step-by-Step Logic Analysis:

1. Analyze the Given Statement:
The given statement is: "We shall go either for a picnic or for trekking."
In logic, the phrasing "either A or B" represents a disjunction. In standard English and propositional logic, this claim asserts that at least one of the two actions will take place. That is, we will go for a picnic, we will go for trekking, or we might even do both (depending on whether the "either... or" is interpreted as inclusive or exclusive disjunction). Under both interpretations, the statement promises that we will perform at least one of these two activities.

2. Determine What Falsifies the Statement:
To falsify or negate a statement of the form "A or B" (which asserts that at least one event will occur), we must show that neither event occurs.
Symbolically, if the statement is represented as:
$$P\vee T$$
where $P$ represents "going for a picnic" and $T$ represents "going for trekking", the negation (falsification) of this statement according to De Morgan's Laws is:
$$\neg (P\vee T)\equiv \neg P\wedge \neg T$$
This translates to: "We do not go for a picnic AND we do not go for trekking."

3. Evaluate the Given Options:
- "We go for a picnic but not for trekking.": This makes the statement true because one of the activities (picnic) was undertaken.
- "Activities such as picnic and trekking are encouraged by the health authorities.": This is an external statement about health recommendations and does not describe our actual actions, so it cannot falsify our claim.
- "We go for trekking and not for picnic.": This also makes the statement true because one of the activities (trekking) was undertaken.
- "We do not go either for picnic or for trekking.": This means we do neither activity. Since we failed to do both, the claim that we would do at least one of them is completely false.

Therefore, the option that falsifies the claim is indeed "We do not go either for picnic or for trekking."