Represent this truth table

Question

I would like to know how I can represent this truth table using the compound proposition below please? $$(P\lor Q)\to(R\land(S\land T))$$

Answer 1

Note that $$\lnot (P\lor Q)\to(R\land(S\land T)) \equiv (P\lor Q) \lor (R\land S\land T)$$

Technically, for a truth-table, you'll need $2^5$ rows to accommodate all possible truth value assignments. So the task is mainly about doing the grunt work (which is easy, but tedious).

Note that any truth-value assignment in which $P = T, Q=T$, or both $P=T$ and $Q=T$ is true, because only one disjunct of the three I display above is needed to make the statement true. The only cases in which the statement the statement might be false is when $P = F, Q= F$, and any one or more of $R, S, T$ is false.

See: The truth-table generator from the Stanford website. Once there, type in the expression "(P or Q) or (R and S and t)", click enter, and you'll be able to generate the 32 rows (5 columns) of truth value assignments, and the resulting truth or falsity of the proposition under the each row's truth value assignments.

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EDIT: It seems the question the OP intended to ask is how to write the truth-table for $$(P\lor Q)\to(R\land(S\land T)) \equiv \lnot (P\lor Q) \lor (R\land S\land T)\equiv (\lnot P \land \lnot Q) \lor(R \land S \land T)$$

In such case, the proposition is true if and only if [($P = Q = \text{false}$), or ($R = S=T= \text{true})]$.