# WEEK NINE SS2 FURTHER MATHS SECOND TERM LOGICAL REASONING

WEEK NINE

SS2 FURTHER MATHS SECOND TERM

LOGICAL REASONING

CONTENT

(a) Clever System

(b) Introduction to Propositional and Predicate Logical Decision

(c) Introduction to Theorem Proving.

SUB TOPIC: INTELLIGENT SYSTEM

An clever system is any formal or casual system designed to:

1. Handle information gathering
2. Get hold of and course of information
3. Interpret the information with a view to giving reasoned makers for the aim of taking determination or acceptable motion. Clever system can be known as synthetic intelligence.

CLASS ACTIVITY:

1. Outline clever system
2. Checklist three issues clever system is designed for

SUB TOPIC: INTRODUCTION TO PROPOSITIONAL AND PREDICATE LOGICAL RESOLUTION

FUNDAMENTAL DEFINITIONS

1. Proposition: it is a written or verbal declarative sentence which is both true or false.
2. True worth: the true worth of a proposition is the reality or falsity of that proposition. Fact worth is denoted as T whereas false worth is denoted as F.
3. Composition proposition: It is a assertion that’s fashioned from the mixture of two or extra easy assertion. It is usually known as compound or molecular proposition.
4. Propositional connectives: These are phrases or symbols which are used to affix two or extra easy statements to kind composite proposition. The desk under exhibits some propositional connectives.

Connective phrases connective symbols

Not

Or ˅

And ˄

If ……then ⇐ or ⇒

If and provided that ⇔

The desk under exhibits the utilization of propositional connective in compound statements.

On condition that p and q are propositions or statements, then:

Negation

Disjunction p˅q

Conjunction P˄q

Implication p ⇒q

Equivalence p⇔q

LAWS OF THE ALGEBRA OF PROPOSITION

Idempotent Legal guidelines

Commutative Legal guidelines

Associative Legal guidelines

Distributive Legal guidelines

Id Legal guidelines

[mediator_tech]

Complement Legal guidelines

Involution Legal guidelines

Demorgan’s Legal guidelines

PREDICATE LOGICAL RESOLUTION

A predicate logic has all of the attributes of a propositional logic in addition to proposition variables and constants. The a part of a declarative sentence which describes the attributes of an object or location amongst objects is known as predicative.

Instance:

• Audu is an educator.
• Bamidele is an educator.

These two propositions clearly have one frequent attribute which is “being an educator” and it’s known as predicate.

CLASS ACTIVITY:

1. Outline propositional logic
2. Outline predicate logic
3. Give two examples of commutative legislation on logic
4. On condition that p and q are statements such that: p: he’s wholesome, q: he’s neat. Write “he’s unhealthy provided that he’s soiled”.
5. What’s a legitimate argument?

SUB TOPIC: PROVING THEOREM

A theorem is a press release in arithmetic that has been confirmed on the idea of earlier established statements reminiscent of theorems or axioms that are accepted statements. Theorem consists of two elements: speculation and conclusion. One of many methods of proving theorems is by mathematical induction. Mathematical induction precept states that if:

1. When a press release is true for a pure quantity then it should even be true for its successor, .
2. The assertion is true for the the assertion shall be true for each pure quantity

CLASS ACTIVITY:

1. Let P(x) be “x+1>5” outlined on the set N of pure numbers. Decide its reality set.
2. Show by mathematical induction that:

PRACTICE EXERCISE:

1. Clarify the next phrases:
2. Composite proposition
3. True worth desk
4. Clever system
5. Predicate
6. If p and q are statements, interprete the next molecular propositions.
7. b. p˅q c. P˄q d. p ⇒q e. p⇔q
8. Show by mathematical induction that: 2+ 4 + 6 + … 2n = n(n+ 1).
9. Negate every of the next statements:
10. All college students are boarders;
11. Some college students are females.
12. Categorical the next proposition utilizing quantifiers, variables and predicate symbols.
13. All fishes can swim
14. Not all fishes can swim
15. Some males are trustworthy
16. Some politicians are usually not trustworthy
17. There’s a woman who likes oranges however not apples.

ASSIGNMENT

1. State which of the next are proposition:
2. Remi is a robust boy
3. Are you going house?
4. 7 + 8 = 15.
5. What a beautiful day!
6. I’m feeling chilly
7. 3 is a good quantity.
8. Give the negation of every of the next:
9. It’s a sizzling day.
11. 3>5
12. Two sides of a triangle are higher than the third aspect.
13. Type a compound assertion through the use of the connectives ‘ánd’ and ‘or’ within the following:
14. He had his bathtub. He went out to the market.
15. He’s right here. He’s there.
16. Decide the reality worth of every of the next statements:
17. 3 + 3 = 7 or 2 x 3 = 6.
18. Abuja is in Egypt and 6 + 5 = 11
19. With out utilizing reality tables, show that: {(pvq’)^(p’vq’)}vq is a tautology.

KEYWORDS

• LOGIC
• PROPOSITION
• PREDICATE
• TRUTH TABLE
• QUANTIFIERS
• VARIABLES
• MATHEMATICAL INDUCTION
• INTELLIGENT SYSTEM
• THEOREM