Form 4 Chapter 4
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Mathematical Reasoning
4.1 : Quantifiers "All" and "Some"
* Constructing statements using the quantifiers "all" and "some"A quantifiers denotes the number of objects or cases involved in a statement.
(a) "All" refers to each and every object or case that satisfies a certain condition.
(b) "Some" refers to several and not every object or case that satisfies a certain condition.*Determining the truth value of statements that contain the quantifier "all"In statement that contains the quantifier "all",each and every objects is being considered in
the statement.If there is one object (or more) that contradicts the statement,then the statement is false.
*Generalising statements using the quantifier "all"Sometimes a statement can be generalised to cover all cases using the quantifier "all" without changing its truth value.
*Constructing true statements using the quantifier "all" or "some"To construct a true statement based on given objects and their properties :
(a)Use the quantifier "all" if each and every object satisfies the given property.
(b)Use the quantifier "some" if there is one or more objects that contradicts with the given property.
4.2 Operations on Statements
*Changing the truth value of statements using the word "not" or "no"1.The word "not" or "no" can be used to change the truth value of a statement.
2.The process of changing the truth value of a statement using the word "not" or "no" is known as negation.
3.~p represent the negation for statement p.
4.Example:
*Identifying two statement from a compound statement that contains the word "and"1.In compound statement containing the word "and" we can identify two statement.
2.For example,7 is an odd number and 14 is an even number.Is a compound statement that is made up from the following two statement.
statement 1 : 7 is an odd number. statement 2 : 14 is an even number.
*Forming compound statement using the word "and"W can use the word "and" to from a compound statement from two statement.
*Identifying two statement from a compound statement that contains the word "or" In a compound statement containing the word "or" we can identify two statement.
statement 1 : -5 < -2 statement 2 :½ = 0.5
*Forming compound statements using the word "or"We can from compound statement from two statement by using the word "or"
*Truth value of compound statement that contains the word "and"1.When two statements are combined with the word "and" the compound statement formed is :
(a) True,if both statement are true.
(b) False, if one of the statement or both the statement are false.
2.The truth value are summarised in the truth table
2.For example,7 is an odd number and 14 is an even number.Is a compound statement that is made up from the following two statement.
statement 1 : 7 is an odd number. statement 2 : 14 is an even number.
*Forming compound statement using the word "and"W can use the word "and" to from a compound statement from two statement.
*Identifying two statement from a compound statement that contains the word "or" In a compound statement containing the word "or" we can identify two statement.
statement 1 : -5 < -2 statement 2 :½ = 0.5
*Forming compound statements using the word "or"We can from compound statement from two statement by using the word "or"
*Truth value of compound statement that contains the word "and"1.When two statements are combined with the word "and" the compound statement formed is :
(a) True,if both statement are true.
(b) False, if one of the statement or both the statement are false.
2.The truth value are summarised in the truth table
*Truth value of a compound statement that contains the word "or"1.when two statements are combined with the word "or" the compound statement formed is :
(a) true,if one of the statement or both the statement are truth.
(b) false,if both the statement are false.
2.The truth values are summarised in the truth table
(a) true,if one of the statement or both the statement are truth.
(b) false,if both the statement are false.
2.The truth values are summarised in the truth table
4.4 Implication
*Antecedent and consequent of an implicationStatement in the form "if p,then q"is known as an implication.p is the antecedent and q is the consequent.
*Writing two implications from a compound statement containing "if and only if"A compound statement in the form "p if and only if q"is a combination of two implications.
Implication 1 : "if p, then q"
Implication 2 : "if q, then p"
*Constructing implications "if p,then q" and "p if and only if q"Based on the given antecedent and consequent,we can construct a mathematical statement in the form :
(a) "if p,then q"
(b) " p if and only if q"
*The converse of implicationFor the implication "if p,then q",the converse of the implication is "if q,then p".
*Truth value of the converse of an implicationThe converse of an implication is not necessarily true.
4.5 Arguments
*Premises and conclusion of an argument1. An argument consist of collection of statements,which are the premises,followed by another statement,which is the
conclusion of the argument.
*Making a conclusion based on the given premisesArgument Form IPremise I : All A re B
Premise II : C is A
Conclusion : C is B
For example,Premise I : All multiples of 10 has the unit digit 0
Premise II : M is a multiple of 10
Conclusion : M has the unit digit 10
4.6 Deduction and induction
*Reasoning by deduction and induction1.Deduction is the process of making a specific conclusion based on a general statement.
2.Induction is the process of making a general conclusion based on specific cases.
*Making conclusion by deductionThrough reasoning by deduction ,we can make conclusion for a specific case based on a general statement.
*Making generalisations by induction.Through reasoning by induction,we can make generalisation based on the pattern of a numerical sequence.
*Antecedent and consequent of an implicationStatement in the form "if p,then q"is known as an implication.p is the antecedent and q is the consequent.
*Writing two implications from a compound statement containing "if and only if"A compound statement in the form "p if and only if q"is a combination of two implications.
Implication 1 : "if p, then q"
Implication 2 : "if q, then p"
*Constructing implications "if p,then q" and "p if and only if q"Based on the given antecedent and consequent,we can construct a mathematical statement in the form :
(a) "if p,then q"
(b) " p if and only if q"
*The converse of implicationFor the implication "if p,then q",the converse of the implication is "if q,then p".
*Truth value of the converse of an implicationThe converse of an implication is not necessarily true.
4.5 Arguments
*Premises and conclusion of an argument1. An argument consist of collection of statements,which are the premises,followed by another statement,which is the
conclusion of the argument.
*Making a conclusion based on the given premisesArgument Form IPremise I : All A re B
Premise II : C is A
Conclusion : C is B
For example,Premise I : All multiples of 10 has the unit digit 0
Premise II : M is a multiple of 10
Conclusion : M has the unit digit 10
4.6 Deduction and induction
*Reasoning by deduction and induction1.Deduction is the process of making a specific conclusion based on a general statement.
2.Induction is the process of making a general conclusion based on specific cases.
*Making conclusion by deductionThrough reasoning by deduction ,we can make conclusion for a specific case based on a general statement.
*Making generalisations by induction.Through reasoning by induction,we can make generalisation based on the pattern of a numerical sequence.
