Statement & conclusion: Reasoning Ability Notes for Azure Certification

Statement & Conclusion – Reasoning Ability

(Azure Certification – Aptitude Section)

Goal:
Learn how to determine whether a conclusion logically follows from a given statement (or set of statements) and how to identify the correct conclusion in multiple‑choice questions.

1. Simple Explanation

Term	What it means	Example
Statement	A fact or set of facts given in the problem.	“All dogs are mammals.”
Conclusion	A claim that must be supported by the statements.	“Shiva is a mammal.”
Validity	A conclusion is valid if it follows logically from the statements; otherwise it is invalid.	From “All dogs are mammals” we can’t conclude “All cats are mammals.” (invalid).

Key idea: Only necessary conclusions can be chosen. A conclusion must be true for every situation that satisfies the statements. If there is even one scenario that satisfies the statements but makes the conclusion false, the conclusion is invalid.

2. Key Points

#	Concept	Why it matters	How to test it
1	Universal Statements (`All A are B`)	They impose a necessary relationship.	If the conclusion uses the same universal form, check if it is a logical consequence.
2	Particular Statements (`Some A are B`)	They only guarantee existence of a subset.	You can’t infer a universal statement from a particular one.
3	Contrapositive	`All A are B` ⇔ `All non‑B are non‑A`	Useful when the conclusion is phrased in the contrapositive form.
4	Logical Connectives	`AND, OR, NOT, IF … THEN`	Pay attention to them; they change the inference pattern.
5	Negation of Statements	`No A are B` is the same as `All A are NOT B`.	Helps in recognizing “not” conclusions.
6	Counter‑Example Method	Try to construct a scenario that satisfies all statements but violates the conclusion.	If you succeed → conclusion invalid.
7	Universal Quantifiers	“All” → `∀`. “Some” → `∃`.	Translate to logical form for clarity.
8	Transitivity	If `All A are B` and `All B are C`, then `All A are C`.	A powerful inference rule.
9	Substitution	Replace a term with an equivalent term or subset.	Useful when extra information is given.
10	Logical Operators with Sets	Set inclusion, intersection, union.	Visualise with Venn diagrams if helpful.

3. Worked Example

Question

Statements

All students are athletes.

Some athletes are cheaters.

All cheaters are non‑honest.

Which of the following is a valid conclusion?
A) Some students are non‑honest.
B) All athletes are non‑honest.
C) All non‑honest people are students.
D) No student is a cheater.

Step‑by‑Step Solution

Step	Action	Reason
1	Identify universal statements: (1) All students → athletes.	Only universal statements can give us universal conclusions.
2	Identify particular statement: (2) Some athletes are cheaters.	This only tells us existence, not universality.
3	Identify universal statement (3): All cheaters → non‑honest.	Enables transitive inference.
4	Look for a chain: Students → Athletes → Cheaters → Non‑honest.	From (1) and (3) we get “All students are non‑honest”? Not directly because (2) is only partial.
5	Test each option with counter‑example.
	Option A – “Some students are non‑honest.”	Possible: If the student who is a cheater is in the subset of athletes that are cheaters. So A can be true.
	Option B – “All athletes are non‑honest.”	False: Only some athletes are cheaters; others may be honest.
	Option C – “All non‑honest people are students.”	Wrong: Non‑honest can include cheaters who are not students.
	Option D – “No student is a cheater.”	Contradicts (2) if the student is also an athlete and a cheater.
6	Choose the only universally valid conclusion: A.	It follows logically and cannot be refuted by any valid counter‑example.

Answer: A) Some students are non‑honest.

4. Quick Revision Summary

Topic	Quick Check
Universal → Universal	`All A are B` ⇒ `All A are B`.
Universal → Particular	`All A are B` ⇒ `Some A are B`. (True, but weaker).
Particular → Universal	❌ (Invalid).
Contrapositive	`All A are B` ⇔ `All non‑B are non‑A`.
Transitivity	`All A are B` & `All B are C` ⇒ `All A are C`.
Counter‑example	If you can build a scenario where statements hold but conclusion fails → conclusion invalid.
Logical Connectives	`AND`: both must hold; `OR`: at least one; `IF ... THEN`: conditional.
Notation	`∀x (A(x) → B(x))` for “All A are B”.
Common Pitfall	Assuming “some X are Y” implies “all X are Y”.
Speed Tip	Translate statements into simple symbols first; then test conclusions with a tiny set of objects.

Final Thought

When faced with a Statement & Conclusion question, always:

List all statements.
Identify the type (universal or particular).
Translate into logical form if needed.
Check each conclusion