The difference between if (imply) vs if and only if (bi-conditional imply) in logic

1. If vs. if and only if
By
Aurora Computer Studies
(auoracs.lk)
1

2. Overview
• If (A -> B)
• If and only if (A <-> B)
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3. If A, then Q
(A -> Q)
3

4. A -> Q - Meaning
• A -> Q
o If A happens then Q happens
o A implies Q
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A
B
C
D
.
.
P
Q
R
.
.
Means :
Other possibilities :

5. A -> Q - Truth table
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A Q A->Q
T T T
T F F
F T T
F F T
A
B
C
D
.
.
P
Q
R
.
.

6. A -> Q - What it means?
• A -> Q
o Yes, I mean;
 Whenever A happens Q happens (or high
probability of Q happen)
One of the reason for Q is A.
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
A
B
C
D
.
.
P
Q
R
.
.

7. A -> Q - What it does not means?
• A -> Q
o No, I do not mean;
• A is the only reason(condition) for Q to happen
• Whenever Q has happened means A has happened
• If A happens, the only effect is Q happening
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
A
B
C
D
.
.
P
Q
R
.
.

8. A -> Q - Example 1
• If it rains, then the ground is wet
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Rain
B
Watering
D
.
.
P
Wet
R
.
.

9. A -> Q - Example 2
• If you study, you can pass the
assignment
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Study
B
Cheat
D
.
.
P
Pass
R
.
.

10. if and only A, then Q
(A <-> Q)
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11. A <-> Q - Meaning
• A <-> Q
o If and only if A happens then Q happens
o A -> Q and Q -> A
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A
.
P
Q
R
S
Means :
Other possibilities :

12. A <-> Q - Truth table
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A Q A->Q
T T T
T F F
F T F
F F T
A
.
P
Q
R
S

13. A <-> Q - What it means?
• A <-> Q
o Yes, I mean;
 Whenever A happens Q happens
 Only when A happens Q happens
 If Q has happened, then A must have happened
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
A
.
P
Q
R
S

14. A <-> Q - What it does not means?
• A <-> Q
o No, I do not mean;
• If A happens then, only Q happens
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
A
.
P
Q
R
S

15. A <-> Q - Example 1
• If and only if the offence is proved,
then accused is declared guilty
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Prove the
offence
.
P
Guilty
Punish
S

16. A <-> Q - Example 2
• If and only if you understand, then
you can solve this problem
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Understand
.
P
Solve
Happy
S

17. Thank You !
Presented by
Aurora Computer Studies
(auoracs.lk)
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18. "We are what we repeatedly do.
Excellence, then, is not an act,
but a habit."
~ Aristotle.
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19. For more resources visit;
http://www.slideshare.net/auroracslk
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