Containee problem

Question

For queries Q₁ and Q₂, we say Q₁ is containedin Q₂, denoted Q₁ C Q₂, iff Q₁(D) C Q₂(D) for every database D.

The container problern for a fixed Query Q_o is the following decision problem:
Given a query Q, decide whether Q_o C Q.

The containee probletn for a fixed guery Q_o is the following decision problem:
Given a query Q, decide whether Q C Q_o.

Formally prove or disprove the following statements:

(a) For every conjunctive query Q₀, there is a polynomial-time algorithm to decide the container problem for Q₀ and for given conjunctive queries Q.

(b) For every conjunctive query Q₀, there is a polynomial-time algorithm to decide the container problem for Q₀ and for given conjunctive queries Q that can be obtained from Q_o by adding some atoms.

(c) For every conjunctive euery Q_o, there is a polynomial-time algorithm to decide the containee problem for Q₀ and for grven conjunctive queries  Q.

(d) For every flrst-order Query Q₀, there is an algorithm to decide the containee problem for Qo and for given first-order queries Q.

To prove a statement, sketch an algorithm, along with an argument why it is polynomial, if possible. To disprove it, provide an M-hardness or undecidability proof.