WEEK TEN DATE……………
TOPIC: Logical reasing continues CONTENT:
- Connectives; (Disjunction and conjunction)
- Tautology and contradiction
Disjunction: In disjunction two statement can be combined by the use of the connective to the truth table. The truth table technique is used to establish whether or not two logical statement are equivalent. Let p = He is a pastor and q = He is a singer
The above statement can be written as either he is a pastor or he is a singer.
Hence, in logical symbols; the statement can be written as p or q, where or means v i.epvq.
NOTE: the statement Pvq is false when both p and q are the false otherwise pvq is true. The truth table for the above statement is given or presented as:
| p | q | Pvq |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
CONJUNCTION: When the connective and is used to combine two statement thus, we have conjunction. Let p = Lagos is in Nigeria
Let q = 3 is an odd number Thus, the above statement can be combined using the connective “and” as in : Lagos is in Nigeria and 3 is an odd number and it can be written as; p and q, where and is symbolically represented as ˄ i.e ˄ means “and”. Hence, p and q = p˄q.
The above statement can be illustrated using a truth table.
NOTE: the statement p˄q is true when the sub statement p and q are both true otherwise p˄q is False.
| p | q | p˄q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
TAUTOLOGY: A compound statement which is always true irrespective of the truth values of the sub statement is called TUATOLOGY. It is represented as T.
Example: Use the truth table to show that the statement pv~p is a tautology.
| p | ~p | pv~p |
| T | F | T |
| T | F | T |
| F | T | T |
| F | T | T |
From the above table it can be observed that the last column has the truth value T. Hence, the statement is
TAUTOLGY.
CONTRADICTION: A compound statement which is always False irrespective of the truth value of the sub statement is called CONTRADICTION. It is usually denoted by F.
Example: Use the truth table to show that the statement p˄~p is a tautology.
| p | ~p | p˄~p |
| T | F | F |
| T | F | F |
| F | T | F |
| F | T | F |
From the above table it can be observed that the last column has the truth value F. Hence, the statement is
CONTRADICTION.
EVALUATION:
1.Copy and complete the truth table below:
| P | q | r | qvr | ~p˄(qvr) |
| T | T | T | T | |
| T | T | F | ||
| T | F | T | T | |
| T | F | F | F | F |
| F | T | T | ||
| F | T | F | ||
| F | F | T | ||
| F | F | F | F |
2. Use the truth table technique to establish the following results:
(i) p˄q = q˄p
( ii.) pv(q˄r) = (pvq)vr
(iii) {p˄(~pvq)}Vq is a tautology
GENERAL EVALUATION:
- Draw the truth table for ~ (p→ ~q) Using the truth tables, prove that:
- p
˄{(~p˄p)V(~p~˄q)} is a contradiction. - {(pv~q)
˄(~pv~q)}Vq is tautology.
Reading Assignment: F/Maths Project 2 pages 30 Exercise 3 Q 9 and 12
WEEKEND ASSIGNMENT
1. Let p = She is short and q = She is beautiful. Write each of the following in symbolic form using p and q.
(i) She is short and beautiful (ii) She is short and but not beautiful (iii) It is false that she is tall and beautiful (iv) She is neither short nor beautiful.
Use the truth table technique to show that
- p↔q = (p→q)
˄(q→p) - (p˄q)˄~(pvq) is a contradiction.
- (~pv~q)v~ (pvr)v(qvr) is a tautology.