1
GATE CSE 2001
+2
-0.6
Let $$f:\,A\, \to B$$ be a function, and let E and F be subsets of A. Consider the following statements about images.

$$S1:\,f\,\left( {E\, \cup \,F} \right)\, = \,f\left( E \right)\, \cup \,f\,\left( F \right)$$
$$S2:\,f\,\left( {E\, \cap \,F} \right)\, = \,f\left( E \right)\, \cap \,f\,\left( F \right)$$
Which of the following is true about S1 and S2?

A
Only S1 is correct
B
Only S2 is correct
C
Both S2 and S2 are correct
D
None of S1 and S2 is correct
2
GATE CSE 2000
+2
-0.6
Let P(S) denote the power set of a set S. Which of the following is always true?
A
$$P\,(P(S))\, = P\,(S)$$
B
$$P\,(S)\, \cap \,P\,(P\,(S)) = \{ \emptyset \}$$
C
$$P\,(S)\,\, \cap \,\,S = P\,(S)$$
D
$$S\,\, \notin \,P(S)$$
3
GATE CSE 2000
+2
-0.6
A relation R is defined on the set of integers as zRy if f (x + y) is even. Which of the following statements is true?
A
R is not an equivalence relation
B
R is an equivalence relation having 1 equivalence class
C
R is an equivalence relation having 2 equivalence classes
D
R is an equivalence relation having 3 equivalence classes
4
GATE CSE 1999
Subjective
+2
-0

(a) Mr. X claims the following:
If a relation R is both symmetric and transitive, then R is reflexive. For this, Mr. X offers the following proof.

"From xRy, using symmetry we get yRx. Now because R is transitive, xRy and yRx togethrer imply xRx. Therefore, R is reflextive."

Briefly point out the flaw in Mr. X' proof.

(b) Give an example of a relation R which is symmetric and transitive but not reflexive.

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