SECTION 1.4 Equivalent Statements 31
1.4 EQUIVALENT STATEMENTS Textbook Reference Section 3.3, 3.4
CLAST OBJECTIVE
" Determine equivalent and non-equivalent statements
Equivalent Statements are statements that are written differently, but hold the same
logical equivalence.
Case 1: “ If p then q ” has three equivalent statements.
RULE
Statement Equivalent Statement
1) If p then q not p or q
2) If p then q q or not p
3) If p then q If not q then not p
Examples
Statement Equivalent Statement
a) If it is blue, then it is the sky. It is not blue or it is the sky.
b) If it is blue, then it is the sky. It is the sky or it is not blue.
c) If it is blue, then it is the sky. If it is not the sky, then it is not blue.
Case 2: Statements beginning with “ It is not true that… ”
RULE
Statement Equivalent Statement
1) ~ (p and q) not p or not q
2) ~ (p or q) not p and not q
3) ~ (if p then q) p and not q
4) ~ (All p are q) Some p are not q
5) ~ (No p are q) Some p are q
6) ~ (Some p are q) No p are q
7) ~ (Some p are not q) All p are q
Examples
Statement Equivalent Statement
d) It is not true that Peter has an A and
Mary has a C.
Peter does not have a A or Mary does not
have a C.
e) It is not true that the car is red or the
pole is not blue. The car is not red and the pole is blue.
