A statement in mathematical logic is either true or false but not both. The truth values of a statement are T and F, respectively. A table that consists of the truth values of the statements for all possible values that are assigned to the variables that appear in the statement, which is compound in nature, is called a truth table. A true statement is termed a valid statement, whereas a false statement is an invalid statement.
The types of statements in mathematical reasoning are simple and compound statements. A simple statement is any statement whose truth value is independent of another statement. It is also defined as a statement that cannot be reduced into many simpler statements. On the other hand, a compound statement is a combination of 2 or more simpler statements. The validity of statements refers to the process of verifying when the given statement is true and not true.
Validity of Statements with ‘AND’
Consider p and q to be two mathematical statements. In order to show that statements p and q are true, the following steps are followed:
Step 1: Show that statement p is true.
Step 2: Show that statement q is true.
Example:
p: 80 is a multiple of 4.
q: 80 is a multiple of 5.
p and q: 80 is a multiple of 4 and 5. (True)
80 = 2 * 40
80 = 16 * 5
Validity of Statements with ‘OR’
Consider p and q to be two mathematical statements. In order to show that statements p or q are true, the following steps are followed:
Step 1: By assuming p is false, show that statement q is true.
Step 2: By assuming q is false, show that statement p is true.
Example:
p: 24 is a multiple of 4.
q: 24 is a multiple of 7.
p or q is true.
24 = 6 * 4
Validity of Statements with ‘If-then’
Consider p and q to be two mathematical statements. In order to prove that statements if p then q, the following steps are followed:
Step 1: Assume p to be true, then prove that q must be true. (Direct method)
Step 2: Assume q to be false, then prove that p must be false. (Contrapositive method)
Example: If x and y are odd integers, then xy is an odd integer.
p: x and y are odd integers.
q: xy is an integer.
Direct method
Let p be true.
Let x = 2n + 1 and y = 2m + 1
xy = (2n + 1) (2m + 1)
= 4mn + 2n + 2m + 1
= 2 (2mn + n + m) + 1
= 2p + 1 where p = 2mn + n + m, which is also an integer
xy is an odd integer.
Contrapositive method
q is not true (p is true).
xy is an even integer.
xy = 2 (n), where n is an integer.
x is even, or y is even, or both x and y are even integers.
p is not true.
This is a contradiction to the statement that “p is true”.
Validity of Statements with ‘If and Only If’
Consider p and q to be two mathematical statements. In order to prove that statements p if and only if q, the following steps are followed:
Step 1: If p is true, then q is true.
Step 2: If q is true, then p is true.
Example: An integer n is odd if and only if n2 is odd.
p: An integer n is odd.
q: n2 is odd.
Let p be true.
n = 2m + 1
n2 = (2m + 1)2
= 4m2 + 4m + 1
= 2p + 1
n2 is an odd integer which implies that q is true.
Contrapositive: If n is an integer and n2 is odd, then n is odd.
Let p be false.
n = 2m
n2 = 4m2 = 2 (2m2)
n2 is not an odd integer, which implies that q is not true.
Validity of Statements – Video Lesson
Frequently Asked Questions
What do you mean by the validity of a statement?
The validity of a statement means checking its truth values.
What do you mean by a conjunction statement?
A conjunction statement is formed by adding two statements using connector AND. The symbol used is ‘∧’. For example, if statements p and q are joined in a statement, the conjunction will be denoted as p ∧ q. If both p and q are true, then this statement is true, else it is false.
What do you mean by a disjunction statement?
A compound statement formed by joining two statements using the connector OR is a disjunction. The symbol used is ‘∨’. p∨q is true when at least one of p and q is true.
What do you mean by negation?
The negation of a statement is the opposite of the given mathematical statement. Negation is represented by a symbol ~.
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